Let $\mathcal C$ be a sufficiently-complete-and-cocomplete category. Let $C \in \mathcal C$, and let $A \rightarrowtail C \leftarrowtail B$ be subobjects. Let $A \cap B = A \times_C B$ be the intersection of these subobjects. Let $AB = A \amalg_{A \cap B} B$ be the pushout of $A,B$ over this intersection. There is an induced map $AB \to C$.

Question: What conditions on $\mathcal C$ ensure that $AB \to C$ is a monomorphism?

It suffices to suppose that $\mathcal C$ is either abelian or a topos, I think. Does it suffice to assume that $\mathcal C$ is exact?

Let $\mathcal C$ be a sufficiently-complete-and-cocomplete category. Let $C \in \mathcal C$, and let $A \rightarrowtail C \leftarrowtail B$ be subobjects. Let $A \cap B = A \times_C B$ be the intersection of these subobjects. Let $AB = A \amalg_{A \cap B} B$ be the pushout of $A,B$ over this intersection. There is an induced map $AB \to C$.

Question: What conditions on $\mathcal C$ ensure that $AB \to C$ is a monomorphism?

It suffices to suppose that $\mathcal C$ is either abelian or a topos, I think. Does it suffice to assume that $\mathcal C$ is exact?

Does it suffice for the category to be adhesive? (and incidentally is every abelian category adhesive?)