Dear Fernando, I do not know of a discussion of the exact result you state, but much of what you need to prove it is in the appendix to: A Note on Actions of a Monoidal Category G. Janelidze and G.M. Kelly Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61-91. Best wishes, Richard On 12 January 2011 18:50, Fernando Muro <fmuro@us.es> wrote:
Dear colleagues,
I'm looking for a reference where the following fact (that I believe to be clearly true) is discussed:
Let V and C be biclosed monoidal categories. Suppose that V is symmetric and that we have a strong braided monoidal functor z : V --> Z(C) to the center of C in the sense of Joyal-Street. Assume further that the functor z(-) \otimes Y : V --> C has a right adjoint Hom(Y,-) : C --> V for any object Y in C. Then C is a monoidal V-category with Hom objects in V given by this right adjoint.
You may assume that V and C are (co)complete if you wish.
It is easy to construct compositions morphisms, etc. in an elementary way, but verifying all laws is a pain. This is why I'm willing to find a reference.
All the best for the new year,
Fernando Muro
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]