Dear Categories Mailing list, Day's reflection theorem gives us a way to construct biclosed monoidal structures on reflective subcategories of the presheaf category of a pro-monoidal category. Is there a converse 'trace' theorem, allowing us to construct promonoidal categories on dense subcategories of monoidal biclosed categories under reasonable conditions? Is there a useful and usable condition to determine if, when given the data of a monoidal biclosed category B and a full dense subcategory A c B, the trace of B on A exists and is a promonoidal structure on A? It seems that Day's 1974 paper uses something like this to prove lemma 3.1.1, but it seems to use the fact that the monoidal biclosed category B is the functor category [A,V] in an essential way, but I don't see exactly where this is used to show the trace is pro-monoidal, especially since Day claims that it only relies on the existence of V-cotensors in B, the fact that B is biclosed monoidal, and the fact that A is dense and full in B (see proof of 3.1.1). Is the proof incorrect, or am I missing something? I ask this because I am trying to find the precise point of error In Ittay Weiss's thesis extending the Boardman-Vogt tensor product to Dendroidal Sets. If anyone knows the exact point in the proof where it all goes wrong, I'd really like to know. If the strategory of proof in 3.1.1 can be applied without using that B is [A,V], it seems like Weiss's proof should have worked. Thank you for your time and attention! Your humble servant, Harry J. Gindi [For admin and other information see: http://www.mta.ca/~cat-dist/ ]