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

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

If it is unique, how to prove that? Else, is there any counter-example?