Naturality of Yoneda Lemma for Covariant Functors

Theorem
Let $C$ be a locally small category.

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

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

Let $C \times \sqbrk {C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A, F} \mapsto \map {\operatorname {Nat} } {h^A, F}$ 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 $\operatorname{id}_{\sqbrk {C, \mathbf {Set} } }$.

Let $\operatorname{ev} : C \times \sqbrk {C, \mathbf {Set} } \to \mathbf {Set}: \tuple {A, F} \mapsto \map F A$ be the functor evaluation functor.

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

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

Let $\tuple {f, \xi}: \tuple {A, F} \to \tuple {B, G} $ be a morphism in $C \times \sqbrk {C, \mathbf {Set} }$.

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

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

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

We have

and:

Also see

 * Naturality of Yoneda Lemma for Contravariant Functors