Endomorphism

A morphism f: A → A, that is, a morphism with identical source and target objects, is an endomorphism of object A.

An endomorphism that is also an isomorphism, is an automorphism.

A split endomorphism is an idempotent endomorphism f if f admits a decomposition f = h ∘ g with g ∘ h = id.

In particular, the Karoubi envelope of a category splits every idempotent morphism.