$\otimes$-Frobenius functors and exact module categories

We call a tensor functor $F:\mathcal{C}\to\mathcal{D}$ between finite tensor categories $\otimes$-Frobenius if its left and right adjoints are isomorphic as $\mathcal{C}$-bimodule functors. We give several characterizations of this notion -- most notably, $F$ is $\otimes$-Frobenius if and only if the centralizer $Z({}_{F}\!{\mathcal{D}}_{\!F})$ is unimodular. We use them to analyze how actions on module categories behave under pullback along $F$. For perfect functors, we show that twisting a $\mathcal{D}$-module category $\mathcal{M}$ along $F$ preserves exactness, and that pivotality, unimodularity, and sphericality are preserved whenever $F$ is $\otimes$-Frobenius (or, more generally, Frobenius with respect to $\mathcal{M}$). Applications include: (i) explicit criteria for $\otimes$-Frobenius functors arising from bialgebra maps $f\!:\!H'\!\to\!H$ between finite-dimensional Hopf algebras; and (ii) criteria ensuring that objects of internal natural transformations are (symmetric) Frobenius algebras in $Z(\mathcal{C})$. Along the way we show that central tensor functors are Frobenius iff they are $\otimes$-Frobenius and that any tensor functor between separable fusion categories is $\otimes$-Frobenius, answering questions of Flake-Laugwitz-Posur.