Automorphism group

context	$\bf C$ … concrete category
context	$X:\mathrm{Ob}_{\bf C}$
definiendum	$\mathrm{Aut}(X)\equiv\langle\!\langle X\cong X,\circ\rangle\!\rangle$

Obviously, $\circ$ denotes the concatentaion of arrows in $\bf C$ here.

The automorphisms (which for concrete $\bf C$ can always be viewed as functions) equipped with $\circ$ indeed form a group.