> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/10-morphisms/endomorphism.md).

# 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.