Hopf comonads on naturally Frobenius map-monoidales

We study monoidal comonads on a naturally Frobenius map-monoidale $M$ in a monoidal bicategory $\mathcal M$. We regard them as bimonoids in the duoidal hom-category $\mathcal M(M,M)$, and generalize to that setting various conditions distinguishing classical Hopf algebras among bialgebras; in particular, we define a notion of antipode in that context. Assuming the existence of certain conservative functors and the splitting of idempotent 2-cells in $\mathcal M$, we show all these Hopf-like conditions to be equivalent. Our results imply in particular several equivalent characterizations of Hopf algebras in braided monoidal categories, of small groupoids, of Hopf algebroids over commutative base algebras, of weak Hopf algebras, and of Hopf monads in the sense of Brugui\`eres and Virelizier.