I'm looking for a reference for two notions related to symmetric monoidal categories and symmetric monoidal functors. Assume that A and B are symmetric monoidal categories, and denote the product operation on both by \tensor. Assume that L is a lax (symmetric) monoidal functor from A to B. Here are the two closely related notions: 1. Let G be a functor from A to B for which there is a natural transformation L(a_1) \tensor G(a_2) -> G(a_1 tensor a_2) for all a_1, a_2 in A making the obvious diagrams commute (one an associativity diagram and the other a unit diagram). One can think of L as acting on G, or of G as "modular" over L (I'm working in the abelian category context, so it is natural to think in terms of rings and modules). 2. Now suppose that L has a right adjoint R and G has a left adjoint F. Also assume that L is strictly (or is it strongly) monoidal, and that R has the "dual" monoidal structure. There seems to be a natural common source for an action of L on G and of R on F. This source is a natural isomorphism of the form F(L(a) \tensor b) \iso a \tensor F(b) for all a in A and b in B. If the appropriate diagrams commute, then it is easy to derive the actions of L on G and R on F from this isomorphism. What intrigues me most about this situation is the way in which the adjunctions go in the opposite direction. These two situations appear throughout some stuff in algebraic topology that I am working on right now. Is there a reference in the literature where these notions are discussed? Here is a standard (somewhat degenerate) example of this situation. Let h: S -> T be a homomorphism of commutative rings. There is an associated pullback functor from T-Mod to S-mod which plays the role of the functors R and F above. This functor has both left adjoint (playing the role of L) and right adjoint (playing the role of G). The pullback functor R is lax monoidal, and its left adjoint L is strict monoidal. The functor L acts on G, and R = F acts on F via the lax monoidal structure on R. In the examples that show up in my topological research, R and F are distinct, and neither F nor G have monoidal structures. Thanks, Gaunce Lewis