Entwined comodules and contramodules over coalgebras with several objects: Frobenius, separability and Maschke theorems

We study module like objects over categorical quotients of algebras by the action of coalgebras with several objects. These take the form of entwined comodules'' and entwined contramodules’’ over a triple $(\mathcal{C},A,\psi )$, where $A$ is an algebra, $\mathcal{C}$ is a coalgebra with several objects and $\psi$ is a collection of maps that ``entwines’’ $\mathcal{C}$ with $A$. Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.