Let P be a k-linear semisimple abelian rigid monoidal category with finite dimensional (over k) Hom-spaces (for a field k).

By a tensored P-category we mean a P-category which admits tensors with objects in P, i.e. for every two objects X,Y in the category and P in P, there is an object P⊗X with an isomorphism:

[P⊗X,Y]≅[P,[X,Y]].Does every enriched functor F between tensored P-categories preserve tensors? (i.e. the natural induced map P⊗F(X)→F(P⊗X) is isomorphism) What happens if the categories are also abelian and F is exact?

**Answer**

No. The functor “take a vector space to its double dual” is linear but does not preserve tensors with infinite-dimensional vector spaces.

Enriched functors are automatically “lax tensored”. An enriched functor F provides a natural map [X,Y]F→[FX,FY] for any X,Y. Consider setting Y=P⊗X, and study:

id∈[P⊗X,P⊗X]≅[P,[X,P⊗X]]F∘→[P,[FX,F(P⊗X)]]≅[P⊗FX,F(P⊗X)].

This gives a canonical map P⊗FX→F(P⊗X) which is natural in P,X and compatible with associativity an unit data. It just isn’t automatically an isomorphism.

On first reading, I missed the requirement that P be rigid. In that case the answer is Yes, enriched functors are automatically tensored. In general, a-priori-lax constructions become strong when there is enough dualizability. Consider the composition

F(P⊗X)→P⊗P∗⊗F(P⊗X)→P⊗F(P∗⊗P⊗X)→P⊗F(X)

where the first map is the unit between P,P∗, the second is the lax monoidality constructed above, and the third is F of the counit. This should give an inverse to the lax monoidality.

**Attribution***Source : Link , Question Author : Mostafa , Answer Author : Theo Johnson-Freyd*