Let $C$ be a locally small category. Is it true that $$ X \cong_C Y \iff C(A, X) \cong_{\mathsf{Set}} C(A, Y) \iff C(X, A) \cong_{\mathsf{Set}} C(Y, A) $$ where we assume the last two isomorphism are natural in $A$?
Direct implication is pretty obvious, and the inverse should be standard (co)Yoneda.
Am I missing something subtle?
I think there is nothing subtle here (except for Yoneda lemma, of course).
A consequence of Yoneda lemma is that the Yoneda embeddings $X \mapsto C(-, X), \; X \mapsto C(X, -)$ are fully faithful functors, and any fully faithful functor "reflects and creates" (in the sense of Exercise 1.5.vi of E. Riehl's book "Categories in context") isomorphisms, which is the meaning of the backwards implications.
(On second thought: Pedantically speaking, in order to justify that the above functors are fully faithful, one needs that the maps on hom sets $C(X,Y) \rightarrow Nat(C(-,X), C(-,Y))$ coming from the Yoneda embedding functor agree with the bijections provided by Yoneda lemma, hence in particular they are bijective themselves. So it requires a check, i.e. one cannot just use the Yoneda lemma as a black box, you need the concrete form of the bijections from Yoneda lemma. But it is rather a pedantic remark.)