Does the bijection of function sets imply bijection of sets?

Question

Let $X,Y$ and $Z$ be sets. Is it true that if the set of functions from $Z$ to $X$ is in bijection to the set of functions from $Z$ to $Y$, then $X$ is in bijection to $Y$? Or are there any subtleties with axioms of set theory? If not, how do you prove it?

Answer 1

There is a bijection between functions from $\mathbb N$ to $\{1,2\}$ and functions from $\mathbb N$ to $\{1,2,3,4\}$.

Answer 2

Your question is not soft, even if you tagged it that way. It is on the other end slightly ambiguously stated. So let me restate it precisely: If, for every set $Z$, the sets $X^Z \cong Y^Z$ , are the sets $X$ and $Y$ isomorphic?

The answer is yes. Just take as $Z$ the singleton set $*$. Then $$Y \cong Y^* \cong X^* \cong X$$