All categorists: I can't respond to Paul Levy's request for sources, but the issue of categorical modules may relate to a general question regarding the necessity of V-enrichment via monoidal categories. So, in a backhanded sense, it could bear on Paul's apparent search for something independent of standard V-enrichment. Please pardon my naivete---my whole concern with this issue began just a few weeks ago during some correspondence with Gabi Lukacs. First, the connection, then the question: Since I am a little out of sympathy with monoidal categories, if I want to enrich the homsets of a category R into objects of some other category C which has a function-valued forgetful functor U on it, I look for a bifunctor r:RxR'--->C (where R' is the opposite category of R) for which the function composite functor U o r is the identity adjunction on R. This adjunction is the bicomposition functor which sends a pair (a,b) to the function z-->azb (which maps between the obvious homsets). If s:SxS'--->C is another such C-enrichment or structure, then a C-structure morphism from r to s is a functor from R to S which satisfies a certain property which is stated in terms of the identity adjunctions which are at issue. It frequently happens that C has a salient C-structure c:CxC'--->C of its own, in which case I am inclined to call a C-structure morphism from r to c an r-module. Take R to be the multiplicative monoid of a small ring, r to be simultaneous left and right multiplication in R and C to be the category of small commutative group homomorphisms to see that a ring is a C-structure and that an r-module is the usual idea of an R-module in this case. I hope that this is what Paul's question is about. I would like to see more information along these lines, myself. My question generally asks for the relationship between what is apparently called V-enrichment and the idea just outlined. I can see that, if I fix an R'-object in the right argument of a C-structure r, I get an r-module (a left regular r-module, in fact), and, by taking left or co- adjoint functors of such modules (if possible), I should get a tensor product concept which should define a monoidal category composition for which the V-enrichment is r. The literature has presumably examined the extent to which this is valid, and I would appreciate being told where. Second, with my ingrained if idiosyncratic prejudices against monoidal categories, I am curious to know if in some impressive sense all (presumably closed?) monoidal categories come about in this way? Are those which don't particularly interesting? Or is the situation the reverse: Every worthwhile monoidal category comes from a C-structure r, but there are important r's which don't provide a monoidal category. Is the full story laid out in a book? Michael Kelley's book is out of print according to Gabi Lukacs. Any help? Pat Donaly