nLab
universal epimorphism

A morphism $f:X\to Y$ is a universal epimorphism if for every morphism $u:V\to Y$ there is a pullback $X{\times }_{Y}V$ and its projection $u^{*}(f):X{\times }_{Y}V\to V$ is an epimorphism.

In particular, setting $u={\mathrm{id}}_Y:Y\to Y$, we see that $f$ itself is an epimorphism.

A morphism $g:X\to Y$ is a universal monomorphism if its opposite $g^{\circ }:Y^{\circ }\to X^{\circ }$ is a universal epimorphism in the opposite category. In particular, it is a monomorphism.