Let $\mathcal A$ be a (small) non-zero abelian category equipped with a monoidal structure $\otimes$ which is right-exact in each variable. (Maybe feel free to assume more if that makes things easier -- for instance maybe assume more colimit-preservation properties, or even that $\otimes$ is symmetric monoidal.)

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. For instance, to answer my question affirmatively, it would suffice to show that $\operatorname{Ind}(\mathcal A^{\mathrm{op}})$ contains a nonzero injective object $I$ admitting an algebra structure (for then you can take $F = \operatorname{Hom}(-,I)$). But according to wikipedia, in general the injective hull of an algebra need not be an algebra, so perhaps the answer to my question is no?