Naturality of Yoneda Lemma for Contravariant Functors

Theorem
Let $C$ be a locally small category.

Let $\mathbf{Set}$ be the category of sets. Let $[C^{\operatorname{op}}, \mathbf{Set}]$ be the contravariant functor category.

Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] $ be the product category.

Let $C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto \operatorname{Nat}(h_A, F)$ be the covariant functor defined as the composition of the hom bifunctor and the product of the opposite of the covariant Yoneda functor $h_-$ and the identity functor $\operatorname{id}_{[C^{\operatorname{op}}, \mathbf{Set}]}$.

Let $\operatorname{ev} : C^{\operatorname{op}} \times [C^{\operatorname{op}}, \mathbf{Set}] \to \mathbf{Set} : (A, F) \mapsto F(A)$ be the contravariant functor evaluation functor.

Then $\Phi_{(A, F)} : \operatorname{Nat}(h_A, F) \to F(A) : \eta \mapsto \eta_A(\operatorname{id}_A)$ defines a natural isomorphism, where $\operatorname{id}_A$ is the identity morphism of $A$.

Also see

 * Naturality of Yoneda Lemma for Covariant Functors