See Ross Street "The formal theory of monads", JPAA 2 (1972) 149-168 for the general definition, in the abstract setting of a 2-category instead of Cat. Actually, there are two obvious generalizations of the TTT definition ("monad functors" and "monad opfunctors"), for the two possible directions of F. Steve Vickers. On 10 Jul 2007, at 14:25, Steven R. Costenoble wrote:
In Toposes, Triples, and Theories, Barr and Wells define a morphism of triples (which, being a student of Peter May, I will call a map of monads) in the context of two monads on a given category C. I have a situation where I have two categories C and D, a monad S on C, a monad T on D, and a functor F: C -> D. There is a fairly obvious generalization of the TTT definition, to say that a map from S to T is a natural transformation FS -> TF making certain diagrams commute. My guess is that someone else noticed this long ago, so I'm looking for references to where this has appeared in the literature. I'm particularly interested in references that include the fact (at least, I'm pretty sure it's a fact) that such maps are in one-to-one correspondence with extensions of F to a functor between the respective Kleisli categories of S and T.
Thanks in advance.
--Steve Costenoble