Let $C$ be a category with limits and $X\rightarrow Y$ a $C$-morphism. Is the induced morphism $X\times_YX\rightarrow X\times X$ a mono?
In the category of sets, $X\times_YX$ is a subset of $X\times X$ and $X\times_YX\rightarrow X\times X$ is a mono. Therefore it should also be true in a topos but I have no idea in a general $C$.