Let be a concrete category.
An endofunctor is a subfunctor of the identity functor if for each object in . That means in particular that the inclusion is a natural transformation. Thus, if is a morphism in , is simply the restriction-corestriction of and .
How to make sense of this in an arbitrary category. Subobject is viewed as an isomorphism class of monomorphisms. So a subfunctor of the identity should be a pair where is an endofunctor and is the natural transformations whose components are monic. The commutative diagram
defines the restriction of and then, by monicity of , determines the arrow uniquely; this is the restriction-corestriction when it exists.
In the particular case if is also an inclusion, . It is clear that
If then by monicity of from the following commutative diagram
we obtain , that is, .
A subfunctor of the identity is called idempotent if is an isomorphism.
In Abelian context, consider additive subfunctors of the identity.
If is left exact, then it is idempotent.
For the proof, start with the exact sequence
By left exactness of ,
Now we decompose
hence and therefore .
Left exactness implies more generally that for any , .
Idempotent kernel functor (for some: radical functor) is left exact subfunctor of the identity satisfying . Identity holds for Jacobson radical, which is in general neither idempotent nor left exact.
Let . The commutative diagram
shows ; therefore . On the other hand,
and if then , that is .
Summarizing, if then .