Normal epimorphism

A morphism having the characteristic property of the natural mapping of a group onto a quotient group or of a ring onto a quotient ring. Let be a category with zero morphisms. A morphism is called a normal epimorphism if every morphism for which it always follows from , , that , can be uniquely represented in the form . The cokernel of any morphism is a normal epimorphism. The converse assertion is false, in general; however, when morphisms in have kernels, then every normal epimorphism is a cokernel. In an Abelian category every epimorphism is normal. The concept of a normal epimorphism is dual to that of a normal monomorphism.