A semifunctor is a homomorphism between semicategories, like a functor is a homomorphism between categories. A semifunctor from a semicategory to a semicategory is a map sending each object to an object and each morphism in to morphism in , such that If is a category, then need not preserve its identity morphisms, but the composition axiom does require that it send them to idempotents in . In Rel, the bicategory of relations, adjoint pairs of relations are equivalent to functions (see principle.