> 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/automorphism.md).

# Automorphism

An endomorphism, which is a morphism whose domain and codomain are the same, that, additionally, is also an isomorphism as well, is an **automorphism**.

An automorphism is a morphism that is both an endomorphism and an isomorphism.

In every category, the automorphisms of an object always form a group, called *the automorphism group of the object*.