Has anyone proved that if you take an "algebra" (actually monoid) object in a monoidal biclosed category that has equalizers and coequalizers, then the category of two-sided modules for that algebra is again a monoidal biclosed category. Mac Lane did this in 1965 when everything is symmetric (the tensor, the algebra and the modules) under the (surely irrelevant) assumption that the original category is also abelian. The fact is certainly true, but writing down the proof would be rather painful.
The question made me think - as perhaps it was meant to - of modules over non-commutative rings, and I wondered whether its scope was unnecessarily restricted in considering just 2-sided modules over a single algebra. We also know that there can be a rich and well-behaved structure for bimodules over two algebras: (A,B)-bimod tensor (B,C)-bimod is (A,C)-bimod B (A,B)-bimod hom (A,C)-bimod is (B,C)-bimod A (A,B)-bimod hom (C,B)-bimod is (C,A)-bimod B Does anyone know how to state the question to cover this more general context? But then what works for algebras ought also to work for algebroids, i.e. enriched categories. Steve Vickers.