  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.
