One often considers regular monomorphisms and regular epimorphisms. In the theory of abelian categories, one often equivalently uses normal monomorphisms and epimorphisms. In general, the concept makes sense in a category with zero morphisms, that is in a category enriched over pointed sets.
Note that a normal monomorphism in is the same as a normal epimorphism in the opposite category , and vice versa.
A category is normal if every monomorphism is normal; it is conormal if every epimorphism is normal, and it is binormal if it is both normal and conormal. Note that is normal if and only if is conormal, and vice versa.
Every normal monomorphism or epimorphism is clearly regular; the converse holds in any Ab-enriched category (the equalizer of and is the kernel of ).
Every abelian category must be binormal; this is one of the axioms in a common definition of abelian category.