Idempotent Monads and *-Functors

For an associative ring R, let P be an R-module with S = EndR(P). C. Menini and A. Orsatti posed the question of when the related functor HomR(P, −) (with left adjoint P ⊗S −) induces an equivalence between a subcategory of R M closed under factor modules and a subcategory of S M closed under submodules. They observed that this is precisely the case if the unit of the adjunction is an epimorphism and the counit is a monomorphism. A module P inducing these properties is called a ⋆-module. The purpose of this paper is to consider the corresponding question for a functor G : B → A between arbitrary categories. We call G a ⋆-functor if it has a left adjoint F : A → B such that the unit of the adjunction is an extremal epimorphism and the counit is an extremal monomorphism. In this case (F, G) is an idempotent pair of functors and induces an equivalence between the category AGF of modules for the monad GF and the category B F G of comodules for the comonad F G. Moreover, B F G = Fix(F G) is closed under factor objects in B, AGF = Fix(GF) is closed under subobjects in A.