  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 $\mathscr C$.

Category Theory