Equivariant Eilenberg-Watts theorem for module coalgebras

For coalgebras $C$ and $D$, Takeuchi proved that the category of linear functors from $\mathfrak{M}^C$ to $\mathfrak{M}^D$ preserving small coproducts is equivalent to the category of $C$-$D$-bicomodules, where $\mathfrak{M}^C$ for a coalgebra $C$ means the category of right $C$-comodules. We formulate and prove an equivariant version of this result for module coalgebras over a bialgebra. As an application, for a bialgebra $H$, we establish an equivalence of the 2-category of a particular class of module categories over the monoidal category $\mathfrak{M}^H$ and the 2-category of a particular class of module categories over the monoidal category ${}_H\mathfrak{M}$ of left $H$-modules.