The First Yoneda Lemma
Concept
The Yoneda lemma states that every natural transformation η:Mor(a,−)⇒F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \eta : \text{Mor}(a, -) \To F is of the form ηb(f)=F(f)(z) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \eta_b(f) = F(f)(z) 1 where z=ηa(1a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } z = \eta_a(1_a).
One may argue that this is at least vaguely intuitive, as the only “natural” (in the sense of naturality) way to transform an a→thing %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a \to \text{thing} into an F(thing) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F(\text{thing}) is if you already have a z∈F(a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } z \in F(a) and just do F(f)(z) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F(f)(z).
Futher, there are no restrictions on what value z %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } z can take in F(a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F(a); for every choice z0∈F(a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } z_0 \in F(a), the construction ηb(f):=F(f)(z0) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \eta_b(f) := F(f)(z_0) forms a natural transformation Mor(a,−)⇒F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{Mor}(a, -) \To F. Together this tells us that the data of a natural transformation Mor(a,−)⇒F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{Mor}(a, -) \To F is the same as the data of an element of F(a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F(a); that is, Nat(Mor(a,−),F) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{Nat}(\text{Mor}(a, -), F) and F(a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F(a) are in bijection.
Formal Statement
Let C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } C be locally small and let F:C→Set %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F : C \to \cat{Set}. The following is a family of bijections. Nat(Mor(a,−),F)≅F(a)η↦ηa(1a)(η given by ηb(f)=F(f)(z))↤z %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align*} \text{Nat}(\text{Mor}(a, -), F) &\cong F(a) \\ \eta &\mapsto \eta_a(1_a) \\ ( \eta \t{{ \text{given} \text{by} }} \eta_b(f) = F(f)(z) ) &\mapsfrom z \end{align*} This family is natural in both F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F and a %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a2.
By “natural in a %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a" we mean that we can take the expression on the left side of the ≅ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cong Nat(Mor(a,−),F) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{Nat}(\text{Mor}(a, -), F) and replace a %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } a with a parameter to obtain an object-mapping F:C→Set %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \fk F : C \to \cat{Set}, F(a):=Nat(Mor(a,−),F) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \fk F(a) := \text{Nat}(\text{Mor}(a, -), F) and we can do the same on the right side to obtain an object-mapping G:C→Set %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \fk G : C \to \cat{Set} G(a):=F(a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \fk G(a) := F(a) and if we upgrade F,G %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \fk F, \fk G to be functors in the canonical way F(f):=exercise for the readerG(f):=F(f) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align*} \fk F(f) &:= \t{{\text{exercise} \text{for} \text{the} \text{reader}}} \\ \fk G(f) &:= F(f) \end{align*} then our family of bijections in question will form a natural bijection F≅G %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \fk F \cong \fk G. “Natural in F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F" works analogously. Note that this explanation might be brushing size issues under the rug. See Riehl for details.
Here Nat(−,−) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \text{Nat}(-, -) gives the set of natural transformations from one functor to another.
Proof (by Discovery)
Consider a single natural transformation η:Mor(a,−)⇒F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\eta : \text{Mor}(\a, -) \Rightarrow F (perhaps a representation of F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}F?). Naturality of η %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\eta tells us that for each f:x→y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}f: x \to y in C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}C the following square commutes:
That is, (f∘−) ; ⁡ηy=ηx ; ⁡F(f) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}(f \circ -) \then \eta_y = \eta_x \then F(f) Equivalently, by applying both sides to an arbitrary g %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}g, ηy(f∘g)=F(f)(ηx(g))(∀g∈Mor(a,x)) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\eta_y(f \circ g) = F(f)(\eta_x(g)) \hspace{30pt} (\forall g \in \mathrm{Mor}(a,x)) Here’s an observation. If we were to fix g %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}g to be an identity, then since f∘g=f %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}f \circ g = f this equation would tell us the value of ηy(f) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\eta_y(f) for every f %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}f. That is, it would characterize the entirety of η %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\eta! To be able to choose g∈Mor(a,x) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}g \in \text{Mor}(\a, x) to be an identity we need that x=a %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}x = \a. So, reapply naturality specifically to morphisms f:a→y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}f : \a \to y in C %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}C (rather than f:x→y %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}f : x \to y), producing for each f %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}f the square
Pushing id∈Mor(a,a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\text{id} \in \text{Mor}(\a, \a) through both paths of the square gives us that ηy(f)=F(f)(ηa(ida))(⋆) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \eta_y(f) = F(f)(\eta_\a (\text{id}_\a)) \mainEquationTag Hence we have entirely determined η %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \eta, as intended. η %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \eta is determined by the single value ηa(ida) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \eta_\a(\text{id}_\a). Further, any choice of z0∈F(a) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } z_0 \in F(a) can be mapped back to a natural transformation μ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mu so that the value of interest μa(ida) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mu_\a(\text{id}_\a) is equal to z0 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } z_0. We do this by fixing μa(ida)=z0 %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mu_a(\text{id}_a) = z_0 (as desired) and inducing the rest of μ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mu via (⋆) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mainEquationRef. Explicitly, μx(f:x→y)={z0x=a and f=idaF(f)(z0)otherwise %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mu_x(f : x \to y) = \begin{cases} z_0 & x = a \t{ and } f = \mathrm{id}_a \\ F(f)(z_0) & \t{otherwise} \end{cases} This simplifies to just μx(f)=F(f)(z0) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mu_x(f) = F(f)(z_0) I claim without proof that μ %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \mu is indeed natural. So we have a bijection Nat(Mor(a,−),F)≅F(a)η↦ηa(ida)(μx given by μx(f)=F(f)(z))↤z %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \begin{align*} \text{Nat}(\text{Mor}(\a,-), F) &\cong F(\a) \\ \eta &\mapsto \eta_\a(\text{id}_\a) \\ (\mu_x \t{ \text{given} \text{by} } \mu_x(f) = F(f)(z)) &\mapsfrom z \end{align*} for every a %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \a. The Yoneda lemma says that these bijections form a natural isomorphism, natural in both F %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } F and a %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \a.
Consequences
If Mor(c,−)≅Mor(d,−) %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } \text{Mor}(c,-) \cong \text{Mor}(d,-) then c≅d %% general %% % shorthands \newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } % norm/magnitude (REMOVE) \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } % magnitude \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } % short mag \newcommand{\card}{ \t{cd} } % cardinality \newcommand{\dcup}{ \sqcup } % disjoint untion \newcommand{\tup}[1]{ \langle {#1} \rangle } % tuples \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\To}{ \Rightarrow } % draw a box outlining some math \newcommand{\box}[1]{ \fbox{$ #1 $} } % f \onall X = { f(x) : x ∈ X } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } % shorthands: various brackets \newcommand{\tpar}[1]{ \left( {#1} \right) } % "tall parens" \newcommand{\tbrak}[1]{ \left[ {#1} \right] } % "tall brackets" \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } % "tall braces" % reverse \mapsto (FIXME: make better) %\newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\mapsfrom}{ \mathrel{↤} } % reverse-order composition \newcommand{\then}{ \operatorname{\ ;\ } } % Like f' represents "f after modification", \pre{f} % represents "f before modification" % TODO: remove this? \newcommand{\pre}[1]{{ \small `{#1} }} % hook arrows \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } % integration d % derivatives: use {\ddn n x y} for (dy/dx) \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } % nth derivative \newcommand{\dd}{ \ddn{} } % first derivative \newcommand{\d}{ \dd{} } % first derivative (no numerator) \newcommand{\dn}[1]{ \ddn{#1}{} } % nth derivative (no numerator) % derivatives: use {\D n x y} for (∂_x y) \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } % no power \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } % first integral %% category theory %% % category names \newcommand{\cat}[1]{{ \sf{#1} }} % yoneda embedding \newcommand{\yo}{よ} % extra long right-arrows \newcommand{\X}{-\!\!\!-\!\!\!} \newcommand{\xlongrightarrow}{ \mathop{ \, \X\longrightarrow \, } } \newcommand{\xxlongrightarrow}{ \mathop{ \, \X\X\longrightarrow \, } } \newcommand{\xxxlongrightarrow}{ \mathop{ \, \X\X\X\longrightarrow \, } } \newcommand{\takenby}[1]{ \overset{#1}{\rightarrow} } \newcommand{\longtakenby}[1]{ \overset{#1}{\longrightarrow} } \newcommand{\xlongtakenby}[1]{ \overset{#1}{\xlongrightarrow} } \newcommand{\xxlongtakenby}[1]{ \overset{#1}{\xxlongrightarrow} } \newcommand{\xxxlongtakenby}[1]{ \overset{#1}{\xxxlongrightarrow} } % represents an anonymous parameter % eg. $f(\apar)$ usually denotes the function $x \mapsto f(x)$ % TODO: remove this? \newcommand{\apar}{ {-} } %% computability %% % turing machines \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \newcommand{\a}{{\mathbf a}}\newcommand\mainEquationRef{{ (\star) }} \newcommand\mainEquationTag{ \tag{$\star$} } c \cong d