Statement: If
κ is a
natural transformation Mor(c,−)⇒Mor(d,−), then exists some
h so that
κx=(−∘h) for every
x.
Proof. Take
κ. Consider the
bijection produced by Yoneda
Φ:Nat(Mor(c,−),Mor(d,−))≅Mor(d,c)
Know that
κ=Φ−1(Φ(κ)); expanding this entails that
κx(f)=Mor(d,−)(f)(κc(1c)) for every
f:c→x. The
action of
Mor(d,−) on
morphisms is post-composition, so this further becomes
κx(f)=f∘κc(1c). Then letting
h=κc(1c):d→c we have
κx=(−∘h):Mor(c,x)→Mor(d,x)
as promised