Idempotent monads and $\star$-functors

For an associative ring $R$, let $P$ be an $R$-module with $S={\text{End}}_R(P)$. C.
Menini and A. Orsatti posed the question of when the related functor ${\text{Hom}}_R(P,-)$ (with left adjoint $P{\text{ot}}_S-$) induces an equivalence between a subcategory of ${}_R\text{M}$ closed under factor modules and a subcategory of ${}_S\text{M}$ 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 $\star$-module. The purpose of this paper is to consider the corresponding question for a functor $G:\text{B}\to \text{A}$ between arbitrary categories. We call $G$ a {\em $\star$-functor} if it has a left adjoint $F:\text{A}\to \text{B}$ such that the unit of the adjunction is an {\em extremal epimorphism} and the counit is an {\em extremal monomorphism}. In this case $(F,G)$ is an idempotent pair of functors and induces an equivalence between the category ${\text{A}}_{GF}$ of modules for the monad $GF$ and the category ${\text{B}}^{FG}$ of comodules for the comonad $FG$. Moreover, ${\text{B}}^{FG}=\text{Fix}(FG)$ is closed under factor objects in $\text{B}$, ${\text{A}}_{GF}=\text{Fix}(GF)$ is closed under subobjects in $\text{A}$.