Dear Tony, As you say, this depends on how the monoidal structures on the functor categories are defined. The only sensible way I know of which uses the given structures on A and B is by Day convolution, and then your result will hold if F itself is strong monoidal. See the Day-Street paper "Kan extensions along promonoidal functors" in volume 1 of TAC. Regards, Steve Lack. On 14/02/09 12:47 AM, "Tony Meman" <tonymeman1@googlemail.com> wrote:

Dear category theorists, I have a question concerning enriched left Kan-extensions.

My situation is the following: V is a complete and cocomplete symmetric monoidal closed category, A and B two small V categories and F:A-->B a V-functor. Via the V-left-Kan extension one gets a V-adjunction Lan_F: V-Fun(A,V)<-->V-Fun(B,V):F* where F* denotes the precomposition with F.

Moreover, the V-category V-Fun(A,V) and the V-category V-Fun(B,V) are equipped with a symmetric V-monoidal structure respectively. Is it known, under which conditions the adjunction (Lan_F,F*) is actually a monoidal adjunction? Surely, it must have something to do with F: I suppose that F have to be a symmetric monoidal functor with respect to a symmetric V-monoidal structure on and A and B. The V-monoidal structures on A and B also should have something to do with the V-monoidal structure on V-Fun(A,V) and V-Fun(B,V).

Does anyone know a reference for this situation?

Thank you in advance for any help. Tony