The purpose of this question is to find some general conditions on a category $\mathcal C$ such that the following result is provable.
Consider two monomorphisms $i : A \hookrightarrow X$ and $j : B \hookrightarrow X$, with trivial intersection in the sense that they have a pullback which is an initial object. Suppose also we have a coproduct $A \sqcup B$.
I would like to know some additional assumptions on $\mathcal C$ that allow me to conclude that the universal morphism $A \sqcup B \to X$ is also a monomorphism.
In particular, these conditions should allow for the case $C = \mathsf{Vect}$, and otherwise "the more general the better".
