The following conditions on a category are equivalent. When they are satisfied, we say that is adhesive.
has pullbacks and pushouts of monomorphisms, and in any cubical diagram:
if is a monomorphism, the bottom square is a pushout, and the left and back faces are pullbacks, then the top face is a pushout if and only if the front and right face are pullbacks. In other words, pushouts of monomorphisms are van Kampen colimits.
In an adhesive category, the pushout of a monomorphism is again a monomorphism.
Any topos is adhesive (Lack-Sobocisnki). For Grothendieck toposes this is easy, because is adhesive and adhesivity is a condition on colimits and finite limits, hence preserved by functor categories and left-exact localizations. For elementary toposes it is a theorem of Lack and Sobocinski.
The fact that monomorphisms are stable under pushouts in toposes plays a central role for Cisinski model structures such as notably the standard model structure on simplicial sets, where the monomorphisms are cofibrations and as such required to be closed under pushout (in particular).