It is analogous to the dual vector space of a vector space: a finite-dimensional vector space "V" and its dual vector space "V*" are not naturally isomorphic, but the endomorphism algebra (matrix algebra) of one is isomorphic to the opposite of the endomorphism algebra of the other: formula_2 via the transpose. Similarly, a group "G" and its dual group formula_3 are not in general isomorphic, but their endomorphism rings are opposite to each other: formula_4. More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see categorical considerations.