There has been some further discussion between Mike Barr and me on modules between monoids in a complete, cocomplete, closed monoidal category. Mike responded:
If I have understood your theorem, it is claimed only for the case that the original category is braided. I was interested in the genuinely asymmetric case, so this doesn't apply. Although your arguments are probably valid in that case, at a guess. Have I missed something?
So I said: It looks as though **I** missed something. I didn't include the older fact well known to enriched category theorists: even without any symmetry or braiding, V-Mod is a bicategory in which all right extensions and right liftings exist (I don't like the word "biclosed"; I use "left closed", "right closed" and "closed" for both when dealing with a monoidal category). In particular, for any V-category A, each hom category V-Mod(A,A) is a closed monoidal category under composition (= tensor product) of modules (as before, I don't say "bimodules" either). Even more particularly you can take A to be a monoid in V. For this there ARE references. I believe Benabou, in his Louvain-la-neuve notes "Les Distributeurs" Rapport No 33 jan 1973, only looked at the case of V symmetric. Same is true of Bill Lawvere's "Metric spaces" paper. But it is not hard to generalise. My paper Enriched categories and cohomology, Quaestiones Math. 6 (1983) 265-283; MR85e:18007 does the non-symmetric case. But it does it more generally for V a bicategory (without commutativity you might as well have "several objects"). Other papers which build on this are: Cauchy characterization of enriched categories, Rendiconti del Seminario Matematico e Fisico di Milano 51 (1981) 217-233; MR85e:18006. (with R. Betti, A. Carboni, and R. Walters) Variation through enrichment, J. Pure Appl. Algebra 29 (1983) 109-127; MR85e:18005. (with A. Carboni, S. Johnson and D. Verity) Modulated bicategories, J. Pure Appl. Algebra 94 (1994) 229-282. Gordon and Power have also done Gabriel-Ulmer duality for W-Mod where W is a decent bicategory (homs locally presentable). JPAA? The recent thing with Brian Day looks at when V-Mod itself is "monoidal", a natural step beyond the above references & studied by Carboni, Walters et al in the case where V is a poset (or W is locally a poset). Best wishes, Ross