More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f : X → → X is an automorphism if there is a morphism g : X → → X such that g ∘ ∘ f = f ∘ ∘ g = id X , where id X is the identity morphism of X. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the identity function, and is often called the trivial automorphism