Adjunctions

Definition

An adjunction consists of two functors $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F : C \leftrightarrows D : G$ alongside a bijection $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } D(F(c), d) \cong C(c, G(d))$$ for all $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c, d$ which is natural in both $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } c$ and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } d$. For this we right $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F \dashv G$ and say that $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } F$ is left-adjoint to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } G$.

Example

Typically free and forgetful functors are adjoint. For instance, let $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cl U : \cat{Grp} \to \cat{Set}$ be the forgetful functor and let $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cl F : \cat{Set} \to \cat{Grp}$ be the free functor taking a set to the free group over that set. For a set $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X$, an element $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } w \in \cl F(X)$ is given by a formal ‘word’ $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } w = w_1^{e_1}\ w_2^{e_2}\ \cdots\ w_k^{e_k}$$ where each $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } w_i$ is an element of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X$ and each $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } e_j$ is an integer and no two adjacent $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } w_i$ are the same elements of $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X$. A homomorphism $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \varphi : \cl F(X) \to H$ out of a free group then consists of choosing some assignment $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h(x) \in H$ for each $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } x \in X$; the value of all words in $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cl F(X)$ follow from this assignment: $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \begin{align*} \varphi(w_1^{e_1}\ w_2^{e_2}\ \cdots\ w_k^{e_k}) &= \varphi(w_1)^{e_1}\ \varphi(w_2)^{e_2}\ \cdots\ \varphi(w_k)^{e_k} \\ &= h(w_1)^{e_1}\ h(w_2)^{e_2}\ \cdots\ h(w_k)^{e_k} \end{align*}$$ That is, a group homomorphism $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \varphi$ contains exactly the same information as an assignment $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } h : X \to \cl U(H)$. This softly-phrased fact strengthens into the existence of a bijection $$\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cat{Grp}(\cl F(X), H) \cong \cat{Set}(X, \cl U(H))$$ which is natural in both $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } X$ and $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } H$. In other words, $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cl F$ is left-adjoint to $\newcommand{\cl}[1]{ \mathcal{#1} } \newcommand{\sc}[1]{ \mathscr{#1} } \newcommand{\bb}[1]{ \mathbb{#1} } \newcommand{\fk}[1]{ \mathfrak{#1} } \renewcommand{\bf}[1]{ \mathbf{#1} } \renewcommand{\sf}[1]{ \mathsf{#1} } \renewcommand{\rm}[1]{ \mathrm{#1} } \newcommand{\floor}[1]{ { \lfloor {#1} \rfloor } } \newcommand{\ceil}[1]{ { \lceil {#1} \rceil } } \newcommand{\ol}[1]{ \overline{#1} } \newcommand{\t}[1]{ \text{#1} } \newcommand{\norm}[1]{ { \lvert {#1} \rvert } } \newcommand{\mag}[1]{ { \left\lvert {#1} \right\rvert } } \newcommand{\smag}[1]{ { \lvert {#1} \rvert } } \newcommand{\card}{ \t{cd} } \newcommand{\dcup}{ \sqcup } \newcommand{\tup}[1]{ \langle {#1} \rangle } \newcommand{\tl}{ \tilde } \newcommand{\wt}{ \widetilde } \newcommand{\onall}[1]{ { \llbracket {#1} \rrbracket } } \newcommand{\tpar}[1]{ \left( {#1} \right) } \newcommand{\tbrak}[1]{ \left[ {#1} \right] } \newcommand{\tbrac}[1]{ \left\{ {#1} \right\} } \newcommand{\mapsfrom}{ \mathop{\leftarrow\!\mid} } \newcommand{\then}{ \operatorname{\ ;\ } } \newcommand{\pre}[1]{{ \small `{#1} }} \newcommand{\injects}{ \hookrightarrow } \newcommand{\embeds}{ \hookrightarrow } \newcommand{\surjects}{ \twoheadrightarrow } \newcommand{\projects}{ \twoheadrightarrow } \newcommand{\id}{ \,\mathrm d } \newcommand{\ddn}[3]{ \frac{ {\mathrm d}^{#1} {#2} }{ {\mathrm d} {#3}^{#1} } } \newcommand{\dd}{ \ddn{} } \newcommand{\d}{ \dd{} } \newcommand{\dn}[1]{ \ddn{#1}{} } \newcommand{\Dn}[2]{ \partial^{#1}_{#2} } \newcommand{\D}{ \Dn{} } \newcommand{\ig}[2]{ \int {#2} \, \mathrm d {#1} } \newcommand{\cat}[1]{{ \sf{#1} }} \newcommand{\yo}{よ} \newcommand{\apar}{ {-} } \newcommand{\halts}{ {\downarrow} } \newcommand{\loops}{ {\uparrow} } \cl U$!