In answering [this MO question](https://mathoverflow.net/questions/25588/free-monad-or-monad-defined-from-an-adjunction/118756#118756), 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](https://mathoverflow.net/questions/335/is-every-functor-a-composition-of-adjoint-functors) 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](https://mathoverflow.net/questions/117401/is-every-functor-inducing-a-homotopy-equivalence-a-composition-of-adjoint-functor), the converse fails, so this is not a characterization.



Can we characterize endofunctors which admit a monad structure?
Characterizing endofunctors on various categories which admit a monad structure is the same as characterizing objects in various monoidal categories which have a monoid structure: "=>" If $(C,\otimes,\dotsc)$ is a monoidal category and $X \in C$, then $- \otimes X : C \to C$ has a monad structure iff $X$ has a monoid structure. "<=" If $C$ is a category and $T : C \to C$ is a functor, then monad structures on $T$ are precisely the monoid structures of $T$ in $(\mathrm{End}(C),\circ,\dotsc)$.


And as a topologist you probably know that it is hard to characterize monoid objects in $\mathrm{hTop}\_\*$ or $\mathrm{hTop}^{\mathrm{op}}\_\\*$ without any further requirements.


But I even doubt that there is any characterization of those endofunctors of $\mathsf{Set}$ which admit a monad structure. I only know of the following necessary condition: The endofunctor preserves epi-mono factorizations up to isomorphism (see Linton, *Coequalizers in categories of algebras*).


