I've asked something similar before, here. But I didn't quite understand their reasoning. So I'm breaking the problem down. First of all, how is $X$ determined?
By yoneda $\text{Hom}_{C^{\wedge}}(h_C(X), F) \simeq F(X)$ so if there is a natural isomorphism $h_C(X) \simeq F$, then ?
I'm sorry! I just don't understand how you get back to $X$ from this.
EDIT. If $h_C(X) \simeq F \simeq h_C(Y)$ then there exists an isomorphism between $h_C(X)$ and $h_C(Y)$ which is induced by $s \in h_C(X,Y)$ by the Yoneda bijection $\psi(s)$. The inverse bijection is $\varphi$ and if $\theta : h_C(X) \to h_C(Y)$ is an isomorphism then since $h_C$ is fully faithful, it reflects isomorphisms, so $\phi(\theta) = s' \in \text{Hom}_C(X,Y)$ is an isomorphism.
So $s