"Functors between monads": what are these really called? 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.  
 A: Pumplün, D. Eine Bemerkung über Monaden und adjungierte Funktoren.
Math. Ann. 185 (1970). 329-337.
also defines a category of monads.
A: As far as I know, the first paper on this was:

Ross Street, The formal theory of monads.  Journal of Pure and Applied Algebra 2 (1972), 149-168.

He called them monad functors.  For the same thing but with the direction of the natural transformation reversed, he called them monad opfunctors.  You can also consider the case where the natural transformation is an isomorphism, or even the identity, but I forget what he called them.
(If Street's wasn't the first paper on this, it was certainly early and very influential.  And it's a beautiful paper.)
In a bid to try to make the terminology of 2-category theory more systematic, I called them lax maps of monads in my book Higher Operads, Higher Categories (CUP, 2004), and similarly colax etc.  I don't know whether anyone else followed suit.
Recently I examined the thesis of one of Street's students, who I think used different terminology from what's in The formal theory of monads.  I forget what it was, but I'll look it up if I remember. 
