Adjointable Abelian Monoidal Functor

Question

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?

Answer 1

Certainly not in general. For instance, let $F$ be the inclusion of the category of finite-dimensional vector spaces (over some fixed field) into the category of all vector spaces.

If you want $\mathcal{C}$ to be complete and cocomplete, here's another counterexample. Let $U$ be a large (i.e., proper-class sized) abelian group, and let $\mathcal{C}$ be category of $U$-graded vector spaces which are $0$ in all but a small set of degrees, equipped with the usual tensor product of graded vector spaces. There is an additive monoidal functor $F:\mathcal{C}\to Vect$ which sends such a graded vector space to the direct sum of its graded pieces. This functor has no adjoint on either side (morally, it "ought" to have a right adjoint that takes a vector space $V$ to the graded vector space that is $V$ in every degree).

I don't know of an example where $\mathcal{C}$ is complete and cocomplete but $F$ fails to have an adjoint on either side because it preserves neither limits nor colimits (as opposed to the previous example which fails to have an adjoint for size reasons). But I don't see any reason that shouldn't be possible. It's certainly easy to come up with examples where $F$ either doesn't preserve limits or doesn't preserve colimits.