nLab universal epimorphism

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

In particular, setting $u = id_Y\colon Y \to Y$, we see that $f$ itself is an epimorphism.

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