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 $x$:

$Aut_C(x) = End_C(x) \cap Iso(C) = Iso_C(x,x)
,$

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