strict monomorphism

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