On Hopf adjunctions, Hopf monads and Frobenius-type properties

Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)\dashv (R,L)$ is a linearly distributive adjunction, if and only if $L\dashv U$ is a Hopf adjunction and $U\dashv R$ is a coHopf adjunction. We give sufficient conditions for a strong monoidal $U$ which is part of a (left) Hopf adjunction $L\dashv U$, to have as right adjoint a twisted version of the left adjoint $L$. In particular, the resulting adjunction will be (left) coHopf. One step further, we prove that if $L$ is precomonadic and $L\mathbf I$ is a Frobenius monoid (where $\mathbf I$ denotes the unit object of the monoidal category), then $L\dashv U\dashv L$ is an ambidextrous adjunction, and $L$ is a Frobenius monoidal functor. We transfer these results to Hopf monads: we show that under suitable exactness assumptions, a Hopf monad $T$ on a monoidal category has a right adjoint which is also a Hopf comonad, if the object $T\mathbf I$ is dualizable as a free $T$-algebra. In particular, if $T\mathbf I$ is a Frobenius monoid in the monoidal category of $T$-algebras and $T$ is of descent type, then $T$ is a Frobenius monad and a Frobenius monoidal functor.