Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors

In this paper we consider a conilpotent coalgebra $C$ over a field $k$. Let $\mathrm{{\rm Y}}\text{\hspace{0.17em}⁣}:C\mathsf{\text{-Comod}}⟶C^{\ast }\mathsf{\text{-Mod}}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^{\ast }$-modules, and let $\mathrm{\Theta }\text{\hspace{0.17em}⁣}:C\mathsf{\text{-Contra}}⟶C^{\ast }\mathsf{\text{-Mod}}$ be the natural forgetful functor. We prove that the functor $\mathrm{{\rm Y}}$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $\mathrm{\Theta }$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space ${\mathrm{Ext}}_C^n(k,k)$ is finite-dimensional for all $n\ge 0$. We call such coalgebras “weakly finitely Koszul”.