Let $\mathcal A$ be a (small) abelian category equipped with a monoidal structure $\otimes$ which is additive and right-exact in each variable.

**Question:** Does there exist a functor $F: \mathcal A \to Ab$ which is nonzero, exact, and lax monoidal?

I think something like this comes up working over a field in the theory of Tannakian categories. But I'm interested in not working over a field. Moreover, I only want the functor to be lax monoidal -- it need not be strong monoidal. I also don't need the functor to be faithful, just nonzero.

This seems related to asking about algebra structures on injective modules -- it seems in general the injective hull of an algebra need not be an algebra, so perhaps the answer to my question is _no_?