TITLE: "Functors between monads": what are these really called?
QUESTION [22 upvotes]: 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.

REPLY [2 votes]: Pumplün, D. Eine Bemerkung über Monaden und adjungierte Funktoren.
Math. Ann. 185 (1970). 329-337.
also defines a category of monads.