In answering this MO question, the issue was raised of characterizing when a given endofunctor $R:C\to C$ has the form $U\circ F$ where $F:C\to D$ is left adjoint to $U:D\to C$, i.e. which admit a monad structure. Is there an algebraic or purely categorical characterization of such $R$?
We know $F$ has to preserve colimits and $U$ has to preserve limits. I'd be happy with an answer saying $R$ is of the form $U\circ F$ iff $R$ preserves $\langle$ fill in the blank $\rangle$. This seems like it should be known classically (e.g. in Categories for the Working Mathematician), but I never learned it and feel like it would be useful to know.
From this MO question we know that in order to be a composition of adjoints $R$ must be a homotopy equivalence on the nerve $N(C)$. According to this MO question, the converse fails, so this is not a characterization.