Examples
In the category of groups, a monomorphism f from H to G is normal if and only if its image is a normal subgroup of G. In particular, if H is a subgroup of G, then the inclusion map i from H to G is a monomorphism, and will be normal if and only if H is a normal subgroup of G. In fact, this is the origin of the term "normal" for monomorphisms.
On the other hand, every epimorphism in the category of groups is normal (since it is the cokernel of its own kernel), so this category is conormal.
In an abelian category, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel. Thus, abelian categories are always binormal. The category of abelian groups is the fundamental example of an abelian category, and accordingly every subgroup of an abelian group is a normal subgroup.
Read more about this topic: Normal Morphism
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