Let C be a comonoid in some preadditive monoidal category $\mathfrak{C}$, then how can we express the category of C-comodules, in terms of some sort of functor category?
I mean is there a similar expression for $^C\mathscr{M}$ as there is for $_CMod$ (or a monoid $C$) which can be expressed as $Add(C:\mathfrak{C})$?