a small technical detail

Question

I would like to understand what it technically means that

regular monomorphisms are stable under pushouts.

And even yet more trivially what does it mean that $h:A\to B$ is stable under pushouts.

Answer 1

It's not meaningful for a single morphism to be stable under pushouts. A class $\mathcal M$ of morphisms, such as the regular monomorphisms, is said to be stable under pushouts if given a span $z\leftarrow x\to y$ with pushout $z\to p\leftarrow y$, if the map $z\leftarrow x$ is in $\mathcal M$ then so is the map $p\leftarrow y$.