The Yoneda Lemma

Let $\mathcal{C}$ be a locally small category,
$A$ be an object in $\operatorname{Ob}(\mathcal{C})$,
and $h_A : \mathcal{C} \to \mathbf{Set}$ be the covariant hom-functor defined by:

• $h_A(X) = \mathcal{C}(A, X)$ for any object $X \in \mathcal{C}$.
• $h_A(f)(g) = f \circ g$ for any morphism $f : X \to Y$ and $g \in \mathcal{C}(A, X)$.

Let $F : \mathcal{C} \to \mathbf{Set}$ be an arbitrary covariant functor.

Prove that there exists a bijection:
$$\Phi : \operatorname{Nat}(h_A, F) \to F(A)$$which is natural in both $A$ and $F$.