Adjunctions between Eilenberg-Moore categories and a PBW-type theorem

Recently, Dotsenko and Tamaroff have shown that a morphism of $T\longrightarrow S$ of monads over a category $\mathscr C$ satisfies the PBW-property if and only if it makes $S$ into a free right $T$-module. We consider an adjunction $\Psi=(G,F)$ between categories $\mathscr C$, $\mathscr D$, a monad $S$ on $\mathscr C$ and a monad $T$ on $\mathscr D$. We show that a morphism $\phi:(\mathscr C,S)\longrightarrow (\mathscr D,T)$ that is well behaved with respect to the adjunction $\Psi$ has a PBW-property if and only if it makes $S$ satisfy a certain freeness condition with respect to $T$-modules with values in $\mathscr C$.