An automorphism of an object in a category is an isomorphism . In other words, an automorphism is an endomorphism that is an isomorphism. Given an object , the automorphisms of form a group under composition, the automorphism group of , which is a submonoid of the endomorphism monoid of : which may be written if the category is understood. Up to equivalence, every group is an automorphism group; see delooping. For any category , there exists a covariant functor from the core to Grp, which maps..