What properties of a category have to be fulfilled such that every diagonal $\Delta:X\to X\times X$ is a monomorphism? Is this true for ''reasonable'' categories?
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Lemma: In general, if $f\circ g$ is a monomorphism, then $g$ is a monomorphism.
Proof: If $g\circ h_1=g\circ h_2$ then $(f\circ g)\circ h_1=(f\circ g)\circ h_2$, and hence, since $f\circ g$ is a monomorphism, $h_1=h_2$. So $g$ is a monomorphism.
Now, the category theory definition of $\Delta$ is that it is the unique function such that $\pi_1\circ \Delta = \pi_2\circ\Delta = 1_X$. And $1_X$ is a monomorphism, so necessarily $\Delta$ is a monomorphism.
So nothing additional is needed, other than the existence of $X\times X$ as a category-theoretic product.