Let $M, N$ be monads of rank $\lambda$, where $\lambda$ is a regular cardinal (I'm primarily interested in the case of finitary monads). Is there a known characterization of functors $\mathrm{Alg}~N \to \mathrm{Alg}~M$ (of course, I mean the Eilenberg-Moore category) coming from monad morphisms $M \to N$ (that is, a morphism of monoids in the monoidal category $\mathrm{End}(\mathrm{Set})$)?



What functors between categories of algebras are induced by morphisms of monads on $\mathrm{Set}$?
The functor $U\_{({-})} : \mathrm{Mnd}(\mathrm{Set})^\circ \to \mathrm{CAT}/\mathrm{Set}$, sending each monad to the forgetful functor from its category of algebras, is fully faithful (Theorem 3 of Frei's [Some remarks on triples](https://link.springer.com/article/10.1007/BF01110118)). Therefore, a functor between categories of algebras for two monads (regardless of rank) corresponds to a monad morphism if and only if the functor commutes with the forgetful functors.


Given such a functor $F : \mathrm{Alg}(N) \to \mathrm{Alg}(M)$, we define a monad morphism $M \to N$ as follows. For each set $x$, we have a free $N$-algebra $Nx$, which is sent by $F$ to an $M$-algebra on $Nx$. Concretely, such an algebra comprises a function $MN x \to N x$ satisfying two laws. Given this function, we precompose by the unit $M\eta\_x : Mx \to MNx$ of $N$, producing a function $Mx \to Nx$. This defines a natural transformation $M \Rightarrow N$, and it can be checked that this is a monad morphism.


In fact, the corresponding statement is true for monads on any category, not just $\mathrm{Set}$.


