A contra variant category-equivalence between categories of right R-modules and left S-modules (all rings have units, all modules are unitary) that contain RR, SS and are closed under submodules and factor modules, is naturally equivalent to a functor Horn (–, U) with a bimodule
are injective cogenerators with S = End UR
and R = End SU, and all modules in are [U-reflexive. Conversely, for any SSUR, Hom(–, U) is a contravariant category equivalence between the categories of [U-reflexive modules, and if U has the properties just stated, then these categories are closed under submodules, factor modules, and finite direct sums and contain RR, UR,
SU. Such a functor will be called a (Morita) duality between R and S induced by U (see (5)).