A functor $F \in C^{\wedge}$ from $C^{op}$ to $\text{Set}$ is representable if there is an isomoprhism $\varphi \in \text{Hom}_{C^{\wedge}}(h_C(X), F)$ for some $X \in C$, we can also write that as $h_C(X) \simeq F$.
I want to prove that $X$ would be determined up to unique isomorphism.
Proof. Suppose that $\varphi: h_C(X) \simeq F$, and $\phi : h_C(Y) \simeq F$, then $\phi \varphi^{-1} : h_C(X) \simeq h_C(Y) \in \text{Hom}_{C^{\wedge}}(h_C(X), h_C(Y)) \simeq h_C(X,Y)$.
How do I complete the proof?
Thanks @Anton
That got me this far:
We have a functor from $C^{op} \to \text{Set}$ given by $h_C(X)$ which is fully faithful. By functoriality, any isomorphism in $C$ is also one under $h_C(X) \circ \text{op}$. Since $h_C$ is fully faithful, we have that an isomorphism in $h_C(X, Y)$ corresponds to one in $\text{Hom}_{C^{\wedge}}(h_C(X), h_C(Y))$ under the Yoneda bijection, $\psi$. Let the "forward" Yoneda bijection be $\varphi$. Then $h_C(X) \simeq F, \ h_C(Y) \simeq F$ gives an isomorphism $h_C(X) \simeq h_C(Y): \theta$. Then $\varphi(\theta) : X \to Y$. We need to show that $\varphi(\theta)$ is unique (the only such isomorphism). Well if $\phi$ were another isomorphism, then $\phi^{-1} \varphi(\theta)$ is an automorphism of $X$. We know that $\text{Hom}_{C^{\wedge}}(h_C(X), h_C(X)) \simeq h_C(X,X)$
But still stuck.
Note that the representing morphisms provided by the Yoneda's lemma satisfy the same relations as the natural transformations they represent, by uniqueness of the representation. This implies that the identity natural transformation is represented by the identity morphism and a composition of natural transformations corresponds to a composition of representing morphisms. Now you can turn an isomorphism between the functors into an isomorphism of representing objects.