Hi - Here are two questions: Suppose you have a category with finite products, say T, and a symmetric monoidal category, say C. Let [T,C] be the category where objects are symmetric monoidal functors from T to C, morphisms are monoidal natural transformations. *1. What structure beyond a mere category does [T,C] automatically get in this sort of situation?* *2. What further structure do we get if C has some particular class of limits or colimits?* I haven't thought about this much. Even if T were just symmetric monoidal, I think [T,C] should get a symmetric monoidal structure due to "pointwise multiplication", just as the set of homomorphisms from one commutative monoid to another becomes a commutative monoid where fg(x) := f(x) g(x) Should [T,C] also have some sort of "comultiplication"? What extra benefits do we get from T being cartesian? Here's why I care: My student Brendan Fong wrote a masters' thesis about Bayesian networks, which he's trying to polish up and publish. In the new improved version, he'll associate to any Bayesian network a category with finite products, say T. This plays the role of a "theory". An assignment of probabilities to random variables consistent with this theory is a symmetric monoidal functor from T to C, where C is some symmetric monoidal category - but not cartesian! - category of probability measure spaces and stochastic maps. So, [T,C] plays the role of the "category of models of T in C". It would be nice to know the properties of [T,C] that follow instantly from what I've said, not reliant on any more detailed information about T and C. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]