domonoid also satisfies López Franco’s naturally Frobenius condition.
In the resulting duoial endohom category, under a minor further assumption about the availability of a conservative forgetful-type functor, it was shown by Böhm–Lack that a bimonoid induces a Hopf monad if and only if it induces a Hopf comonad, if and only if it possesses an antipode (connecting the bimonoid with its “reversed” mate).
This still covers the above examples of weak bialgebras and bialgebroids over a central base (w