Naturality of Yoneda Lemma for Covariant Functors

Theorem
Let $C$ be a locally small category.

Let $\mathbf{Set}$ be the category of sets. Let $[C, \mathbf{Set}]$ be the covariant functor category.

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

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

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

Proof
By the Bijection in Yoneda Lemma for Covariant Functors, $\Phi_{(A, F)}$ is a bijection for all $(A,F)$.

Let $(f, \xi) : (A, F) \to (B, G)$ be a morphism in $C \times [C, \mathbf{Set}]$.

To prove that $\Phi$ is a natural isomorphism, it remains to prove that the following diagram commutes:
 * $\xymatrix{

\operatorname{Nat}(h^A, F) \ar[d]^{\Phi_{(A, F)}} \ar[r]^{\operatorname{Nat}(h^f, \xi)} & \operatorname{Nat}(h^B, G) \ar[d]^{\Phi_{(B, G)}} \\ F(A) \ar[r]^{\operatorname{ev}(f, \xi)}         & G(B) }$

Let $\eta \in \operatorname{Nat}(h^A, F)$.

We have

and

Also see

 * Naturality of Yoneda Lemma for Contravariant Functors