Let $C$ be a category with finite hom-sets. Suppose that $X$ and $Y$ are objects in $C$ such that $C(Z,X)\cong C(Z,Y)$ for any Z (with no naturality condition). For which categories $C$ does it follow that $X \cong Y$? (Of course, it is true for posets).

A somewhat related question is the following.

Let $C$ be a symmetric monoidal closed category. Suppose that $X$ and $Y$ are objects in $C$ such that $[X,Z]\cong [Y,Z]$ for any Z (with no naturality condition). For which categories $C$ does it follow that $X \cong Y$?