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

By a tensored $\cal{P}$-category we mean a $\cal{P}$-category which admits tensors with objects in $\cal{P}$, i.e. for every two objects $X,Y$ in the category and $P$ in $\cal{P}$, there is an object $P\otimes X$ with an isomorphism: $$[P\otimes X,Y] \cong [P,[X,Y]].$$

Does every enriched functor $F$ between tensored $\cal{P}$-categories preserve tensors? (i.e. the natural induced map $P\otimes F(X)\to F(P\otimes X)$ is isomorphism) What happens if the categories are also abelian and $F$ is exact?