I have not seen a name for the structure R. Loader inquires about. It may be a lax version of a structure which has been considered in the context of "quantum topology" (cf. my paper in Topology '90): Let C be a monoidal category (e.g. category with (chosen) products), let X be category equipped with an "action" of C, i.e. a functor a:C x X ----> X, satisfying the "obvious" coherence conditions. X is then a "C-module" and a functor F:X--->X such that for all c in C, x in X such that there is a natural isomorphism from F(a(c,x)) to a(x,F(x)) again satisfying "obvious" coherence conditions is a "C-module functor". In the case in question C = X, tensor = a = cartesian product. I think, however, that one wants a natural transformation, not a natural isomorphism. Best Thoughts, David Yetter