Let be a fixed category.
Consider a morphism in then, for any object in , is the identity map of since for any ,
Given a morphism in , we have that
and so . Similarly, for a morphism we have that,
and so .
Consider morphisms and in , we can define the induced map for by
and we have that
For a morphism in we have that is a monomorphism if and only if is injective for every object in . Suppose is a monomorphism in , then for any object in and any pair of parallel morphisms , we have that
since is a monomorphism. So, is injective for any object in .
Conversely, suppose is injective for any object , then for any pair of parallel morphisms ,
and so
hence, f is a monomorphism in . Similarly, is an epimorphism in if and only if for every object the map is injective.