Frobenius monoidal functors from (co)Hopf adjunctions

Let $U:\mathcal{C}\to \mathcal{D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U⊣R$ is coHopf. Building on the work of Balan, we show that $R$ is separable (resp., special) Frobenius monoidal if and only if $R({\mathbb{1}}_{\mathcal{D}})$ is a separable (resp., special) Frobenius algebra in $\mathcal{C}$. If further, $\mathcal{C},\mathcal{D}$ are pivotal (resp., ribbon) categories and $U$ is a pivotal (resp., braided pivotal) functor, then $R$ is a pivotal (resp., ribbon) functor if and only if $R({\mathbb{1}}_{\mathcal{D}})$ is a symmetric Frobenius algebra in $\mathcal{C}$. As an application, we construct Frobenius monoidal functors going into the Drinfeld center $\mathcal{Z}(\mathcal{C})$, thereby producing Frobenius algebras in it.