strict epimorphism

A morphism $e : A \to B$ is a strict epimorphism if it is the joint coequalizer of all pairs of morphisms $f,g : C \rightrightarrows A$ that it coequalizes. That is, $e$ is an epimorphism, and a morphism $t : A \to T$ factors through $e$ if we have $t \circ f = t \circ g$ for all morphisms $f,g : C \rightrightarrows A$ that satisfy $e \circ f = e \circ g$. In other words, the minimal requirement for a morphism to factor through $e$ is actually sufficient.By the implications below, strict epimorphisms are closely related to effective and regular epimorphisms. Every effective epimorphism is regular and hence strict, and in a category with pullbacks, every strict epimorphism is effective. Thus, in categories with pullbacks, all three mentioned classes of epimorphisms coincide. See also this overview.