ct.category theory - Methods for constructing Frobenius structures

Let ${mathbb F}:({mathcal A},{mathcal E}_{mathcal A})to({mathcal B},{mathcal E}_{mathcal B})$ be an exact functor between exact categories, and suppose ${mathbb F}$ has both a left adjoint ${mathbb F}_lambda$ and a right adjoint ${mathbb F}_rho$. Then the class

${mathcal E} := {Xto Yto Zin{mathcal E}_{mathcal A} | {mathbb F}Xto{mathbb F}Yto{mathbb F}Zin{mathcal E}_{mathcal B}}$

defines another exact structure on ${mathcal A}$.

It is interesting to ask for criteria to decide when this exact structure is Frobenius. One such criterion is the following:

Suppose ${mathcal E}_{mathcal B}$ is the split exact structure, and that for each $Xin{mathcal A}$ the unit $eta_X: Xto{mathbb F}_rho{mathbb F}X$ is an ${mathcal E}_{mathcal A}$-monomorphism, while the counit ${mathbb F}_lambda{mathbb F}Xto X$ is an ${mathcal E}_{mathcal A}$-epimorphism. Then $({mathcal A},{mathcal E})$ has enough projectives and injectives, and the classes ${mathcal P}$/${mathcal I}$ of projective/injective objects in $({mathcal A},{mathcal E})$ are given by

${mathcal P} = {Xin{mathcal A} | Xtext{ is a summand of some }{mathbb F}_lambda , Yin{mathcal B}}$ and

${mathcal I} = {Xin{mathcal A} | Xtext{ is a summand of some }{mathbb F}_rho Y, Yin{mathcal B}}$,

respectively. Consequently, if ${mathbb F}_lambda$ and ${mathbb F}_rho$ have the same image, then $({mathcal A},{mathcal E})$ is Frobenius.

This seems very restrictive, but in fact there are at least two cases I know where it can be applied:

(1) If ${mathcal A}$ is a dg-category, then the forgetful functor ${mathbb F}: text{dg-mod}({mathcal A})totext{gr-mod}({mathcal A})$ fulfills the requirements of the criterion above and thus can be used to construct a Frobenius structure on $text{dg-mod}({mathcal A})$ (for pretriangulated dg-categories ${mathcal A}$, this structure can in turn be restricted to ${mathcal A}$ itself).

(2) If $G$ is a finite group, $H$ is a subgroup, then the fortgetful functor $Gtext{-mod}to Htext{-mod}$ has left adjoint $text{Ind}^G_H$ and right adjoint $text{Coind}^G_H$, and these two functors coincide for $(G:H)<infty$. In this case, the above criterion therefore applies to provide $Gtext{-mod}$ with a Frobenius structure "relative to $H$".

Question

Do you know more criteria for constructing Frobenius structures and situations where they can be applied?

For example, I would be interested in a criterion which can be applied to show that the category of maximal Cohen-Macaulay modules over a Gorenstein ring is Frobenius.