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)⊣(R,L) is a linearly distributive adjunction, if and only if L⊣U is a Hopf adjunction and U⊣R is a coHopf adjunction. We give sufficient conditions for a strong monoidal U which is part of a (left) Hopf adjunction L⊣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 \(L\mathbb {1}\) is a Frobenius monoid (where \(\mathbb {1}\) denotes the unit object of the monoidal category), then L⊣U⊣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 \(T\mathbb {1}\) is dualizable as a free T-algebra. In particular, if \(T\mathbb {1}\) 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.