Weak Frobenius monads and Frobenius bimodules

As observed by Eilenberg and Moore (1965), for a monad $F$ with right adjoint comonad $G$ on any category $\mathbb{A}$, the category of unital $F$-modules $\mathbb{A}_F$ is isomorphic to the category of counital $G$-comodules $\mathbb{A}^G$. The monad $F$ is Frobenius provided we have $F=G$ and then $\mathbb{A}_F\simeq \mathbb{A}^F$. Here we investigate which kind of isomorphisms can be obtained for non-unital monads and non-counital comonads. For this we observe that the mentioned isomorphism is in fact an isomorphisms between $\mathbb{A}_F$ and the category of bimodules $\mathbb{A}^F_F$ subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad $(F,m,\eta)$ and a weak comonad $(F,\delta,\varepsilon)$ satisfying $Fm\cdot \delta F = \delta \cdot m = mF\cdot F\delta$ and $m\cdot F\eta = F\varepsilon\cdot \delta$, the category of compatible $F$-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible $F$-comodules.