Let $C$ be a coalgebra over a commutative ring $R$, if $C$ is cauchy (f.g. and projective), then there is an equivalence of categories between $\operatorname{Comod}(C)$ and the category $\operatorname{Mod}(C^*)$ of modules over the dual algebra $C^*=\operatorname{hom}_R(C,R)$ (See https://mathoverflow.net/questions/94115/when-a-comodule-category-is-equivalent-to-a-module-category )
If $C$ is itself a hopf algebra then this categories are both monoidal with tensor product given by $\otimes_R$. My question: is this equivalence of categories monoidal?