Is the cartesian product of objects in an elementary topos cancellative?

Question

My question is the internalization of this question to an elementary topos $C$.

Is it true that: For objects $X,Y$ and $Z$ in an elementary topos $C$ with $X\times Y\cong X\times Z$, then also $Y\cong Z$?

Answer 1

Consider the elementary topos $\mathsf{Set}$, where we assume the axiom of countable choice. If $X$ and $Y$ are countably infinite sets, and $Z$ is a finite non-empty set, then $X\times Y\cong X\times Z$ but $Y\not\cong Z$.

Answer 2

In addition to Zev's nontrivial counterexample, there is the following trivial one: Consider the case $X = \emptyset$ (initial object). Then $X \times Y$ and $X \times Z$ are initial as well (because in a topos, the initial object is strict and we have the projection morphisms) and thus always isomorphic, irrespective of $Y$ and $Z$.