Equivalence of categories, Gruson-Jensen duality, and applications

For coalgebras $C$ over a field, we study when the categories ${}^C\Mm$ of left $C$-comodules and $\Mm^C$ of right $C$-comodules are symmetric categories, in the sense that there is a duality between the categories of finitely presented unitary left $R$-modules and finitely presented unitary left $L$-modules, where $R$ and $L$ are the functor rings associated to the finitely accessible categories ${}^C\Mm$ and $\Mm^C$.