Abstract We call a tensor functor F: C D between finite tensor categories -Frobenius if its left and right adjoints are isomorphic as C-bimodule functors. We give several characterizations of this notion—most notably, F is -Frobenius if and only if the centralizer Z (₅\!D\!₅) 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 D-module category M along F preserves exactness, and that pivotality, unimodularity, and sphericality are preserved whenever F is -Frobenius (or, more generally, Frobenius with respect to M). Applications include: (i) explicit criteria for -Frobenius functors arising from bialgebra maps f\!: \!H^\! \!H between finite-dimensional Hopf algebras; and (ii) criteria ensuring that objects of internal natural transformations are (symmetric) Frobenius algebras in Z (C). Along the way we show that central tensor functors are Frobenius iff they are -Frobenius and that any tensor functor between separable fusion categories is -Frobenius, answering questions of Flake–Laugwitz–Posur from 18.
Jaklitsch et al. (Thu,) studied this question.