I am wondering if there is only one unique Yoneda isomorphism, that is a natural isomorphism (natural in C and P, that is) between Hom(yC,P) and PC.

The Yoneda lemma says that there exists at least one, and its common proof is constructive, so we have an example. But is it the only one?

If it is unique, how do we prove that? If not, is there any counter-example?