Contravariant Yoneda lemma

Question

I am learning category theory and trying to prove the contravariant Yoneda lemma:

Let $\mathcal C$ be a locally small category and $F:\mathcal C\to\mathsf{Set}$ be a contravariant functor. Fix an object $c\in\mathcal C_0$. Then the function $$ \Phi:\operatorname{Nat}\left(\operatorname{Hom}_\mathcal C(-,c),F\right)\to F(c),\qquad \Phi(\alpha)=\alpha_c(\mathrm{id}_{c}) $$ is a bijection.

I want to prove it using the covariant Yoneda lemma and duality, but I am a little confused. This stackexchange question partially explains the answer, but I think it is sketchy and it does not explain how the natural transformations between contravariant functors correspond to the usual natural transformations. This is exactly what I am confused by.

Can somebody give a fully rigorous proof of this fact? Thank you!

Answer 1

A contravariant functor $C\to D$ is the same thing as a covariant functor $C^{op}\to D.$ So the category of contravariant functors $C\to D$ is the same thing as the category of covariant functors $C^{op}\to D.$ So a natural transformation between two contravariant functors $ C\to D$ is the same thing as a natural transformation between two covariant functors $C^{op}\to D$.

There is no "rigorous proof" to be had here... it is just a matter of definitions.

The covariant Yoneda lemma is $$ \operatorname{Hom}_{\mathrm{Set}^C}(\operatorname{Hom}_C(c,-), F)\cong F(c)$$ for $c\in C$ and $F:C\to \mathrm{Set}$.

Substituting $C$ with $C^{op}$ and using the fact that $\operatorname{Hom_{C^{op}}(c,-) = \operatorname{Hom}_C(-,c)}$ gives $$ \operatorname{Hom}_{\mathrm{Set}^{C^{op}}}(\operatorname{Hom}_C(-,c), F)\cong F(c)$$ for $c\in C$ and $F: C^{op}\to \mathrm{Set}.$ This is the contravariant Yoneda lemma.