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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 $C$.

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up vote 4 down vote accepted

Let $f$ be the given morphism $X \to Y$, and $\pi_1$ and $\pi_2$ be the projections $X \times X \to X$. Then the natural map $X \times_Y X \to X \times X$ is the equalizer of $f \circ \pi_1$ and $f \circ \pi_2$, and hence is a (regular) monomorphism.

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