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.



"Functors between monads": what are these really called?
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. 


