Grothendieck-Verdier functors

We introduce Grothendieck-Verdier functors between Grothendieck-Verdier, or $\ast$-autonomous, categories. Such functors are lax monoidal functors equipped with a morphism expressing compatibility with Grothendieck-Verdier duality. We show that the resulting $2$-category is $2$-equivalent to that of linearly distributive categories with negation and Frobenius linearly distributive functors. We further extend this $2$-equivalence to the braided setting.
We then establish a lifting theorem for Grothendieck-Verdier functors: given a conservative lax monoidal functor from a closed monoidal category $\mathcal{C}$ to a Grothendieck-Verdier category $\mathcal{D}$, we identify additional structure such that the Grothendieck-Verdier structure of $\mathcal{D}$ lifts to $\mathcal{C}$. This structure turns the functor into a Grothendieck-Verdier functor. As applications, 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.