Are two objects equivalent in a category when their hom sets are isomorphic?

Question

For instance, if, for all $a$ in a category $C$, $\hom(a, b)$ is equal to $\hom(a, c)$ when all instances of $c$ are replaced by $b$ in the latter, does this imply equivalence of $b$ and $c$ in $C$? If so, is this a necessary condition for equivalence of objects in $C$?