A monomorphism is a category is said to be normal if it is a kernel (of a morphism). A subobject of an object is normal if any (and hence all) of its representing monomorphisms is normal.

For example, in Grp, the category of groups, the inclusion of a subgroup $H\subseteq G$ into $G$ is normal iff $H$ is a normal subgroup of $G$.

A category is said to be normal if every monic is a kernel. Equivalently, a normal category is a category in which every subobject of every object is normal.

Dually, an epimorphism is conormal if it is a cokernel (of a morphism). A quotient object of an object is conormal if any (and hence all) of its representing epimorphisms is conormal. A category is said to be conormal if every epimorphism is conormal.

The category AbGrp of abelian groups, and more generally, any abelian category, is normal and conormal.