nLab
automorphism

An automorphism of an object $x$ in a category $C$ is an isomorphism $f:x\to x$. In other words, an automorphism is an isomorphism that is an endomorphism.

Given an object $x$, the automorphisms of $x$ form a group under composition, the automorphism group of $x$, which is a submonoid of the endomorphism monoid of $x$:

$${\mathrm{Aut}}_C(x)={\mathrm{End}}_C(x)\cap \mathrm{Iso}(C)={\mathrm{Iso}}_C(x,x),$$Aut_C(x) = End_C(x) \cap Iso(C) = Iso_C(x,x) ,

which may be written $\mathrm{Aut}(x)$ if the category $C$ is understood. Up to equivalence, every group is an automorphism group; see delooping.