Let $(S,\eta,\mu)$ be a monad on a category $C$, and $(T,\eta,\mu)$ a monad on a category $D$. The following kind of gadget is ubiquitous: a functor $F:D\to C$, together with a natural map $\sigma: SF\to FT$, satisfying the identities

- $F\eta = \sigma\circ \eta F$, and
- $\sigma\circ \mu F = F\mu\circ \sigma T\circ S\sigma$.

I might call this a "functor between monads". **Question:** What do people actually call these, and what are the standard references?

Some random examples:

$S$ is the free commutative ring monad on abelian groups, $T$ is the free commutative monoid monad on sets, and $F$ is the free abelian group on a set. (In this case, $\sigma$ is an isomorphism.)

$S=T=J$, the free associative monoid on pointed spaces (where we force the basepoint to be the unit element), with $F=$ loop space. (Enriched category theory provides a lot of examples like this.)

If $F=$ identity functor, we recover the usual notion of morphism between monads.

Some notes:

A functor between monads as above gives you a way to turn a $T$-algebra into an $S$-algebra, so you obtain a functor $F^*: \mathrm{Alg}_T\to \mathrm{Alg}_S$.

You can compose "functors between monads", so there is a category with these as morphisms, and monads as the objects.