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 \((\mathscr C,A,\psi)\), where \(A\) is an algebra, \(\mathscr C\) is a coalgebra with several objects and \(\psi\) is a collection of maps that ``entwines'' \(\mathscr C\) with \(A\). Our objective is to prove Frobenius, separability and Maschke type theorems for functors between categories of entwined comodules and entwined contramodules.