Let $U:\mathcal{ C}\rightarrow\mathcal{ D}$ be a strong monoidal functor between abelian monoidal categories admitting a right adjoint $R$, such that $R$ is exact, faithful and the adjunction $U\dashv R$ 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.
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