Our goal is to show that $\text{Hom}_{C^{\wedge}}(h_C(X), A) \simeq A(X)$.
We first want to show a map from left to right.
The book says:
$$ \text{Hom}_{C^{\wedge}}(h_C(X), A) \to \text{Hom}_{\text{Set}}(\text{Hom}_C(X,X), A) \to A(X) $$
I'm wondering what the first mapping is, and why the subscript changes to $\text{Set}$, and how do they justify that step?
Thanks.