A lifting theorem for Grothendieck-Verdier categories

We identify additional structure on a conservative lax monoidal functor from a closed monoidal category $\mathcal{C}$ to a Grothendieck-Verdier category $\mathcal{D}$, such that the Grothendieck-Verdier structure of $\mathcal{D}$ lifts to $\mathcal{C}$ and the functor becomes Frobenius linearly distributive. As an application, we recover and extend conditions under which modules over Hopf monads and Hopf algebroids inherit Grothendieck-Verdier structures. We also characterize when categories of bimodules, modules, and local modules over (commutative) algebras internal to a Grothendieck-Verdier category admit such structures. Our results apply to quantales, smash product algebras, skew group algebras, and enveloping algebras of Lie-Rinehart algebras. For applications of the lifting theorem, we construct a strict $2$-equivalence between a $2$-category of Grothendieck-Verdier categories and one of linearly distributive categories with negation, and extend this $2$-equivalence to the braided setting.