An [[Automorphism]] is a [[Function|mapping]] $f$ that is an [[Isomorphism]] from a [[Category]] $C$ to that same category. For any [[Algebraic Structure|Algebraic System]], an [[Automorphism]] is defined as a [[Bijective]] [[Homomorphism]]. $\huge \begin{align} f: C \to C \\ f(f^{-1}(x)) \mapsto x\\ \end{align} $ For any [[Category]], there exists the [[Trivial Automorphism]]. ### Automorphism Group For an object $X$ in the [[Category]], the collection of all [[Automorphism|Automorphisms]] form a [[Group]] whose operation is [[Composition]] of [[Morphism|Morphisms]]. The notation for this group is the following: $\huge \op{Aut}_{C}(X) $