Bijection in Yoneda Lemma for Covariant Functors

Theorem
Let $C$ be a locally small category.

Let $\mathbf{Set}$ be the category of sets. Let $F : C \to \mathbf{Set}$ be a covariant functor.

Let $A\in C$ be an object.

Let $\operatorname{id}_A$ be its identity morphism.

Let $h^A = \operatorname{Hom}(A, -)$ be its covariant hom-functor.

The class of natural transformations $\operatorname{Nat}(h^A, F)$ is a small class, and:
 * $\operatorname{Nat}(h^A, F) \to F(A) : \eta \mapsto \eta_A(\operatorname{id}_A)$

and
 * $F(A) \to \operatorname{Nat}(h^A, F) : u \mapsto (X \mapsto (f \mapsto (F(f))(u)))$

are reverse bijections.

Also see

 * Bijection in Yoneda Lemma for Contravariant Functors