When dealing with $\textbf{Set}$ we have that if $f:A\to B$ is a monomorphism, $g:A\to A’$ is an epimorphism, and adding $f’:A’\to B$ we have a commuting triangle, then $f’$ must be a monomorphism. This does not in general hold, despite being intuitive when thinking of monos as keeping all the underlying “parts” of their domains separate, and epis as covering the “parts” of their codomains. This leads to somewhat confusing definitions elsewhere, such as extremal monomorphisms being extremal among all commuting morphisms, not just monomorphisms, where the definition and its alternative coincide in $\textbf{Set}$, and both intuitively do the job they’re supposed to do.
I was wondering if there was a sensible class of monos and epis that behave as above, but constitute all monos and epis in categories nice enough for this to be possible like $\textbf{Top}$. To be more precise, I’m interested in some subclass of monomorphisms such that, in any category $\mathcal{C}$, if $f:A\to B$ belongs to the subclass in $\mathcal{C}$, $g:A\to A’$ belongs to the subclass in $\mathcal{C}^{\textrm{op}}$, and adding $f’:A’\to B$ we have commuting triangle, then $f’$ must belong to the subclass in $\mathcal{C}$.