Heavily separable cowreaths (Q2035816)

Let \(\mathcal{C}\) be a strict monoidal category. The authors define heavily coseparable coalgebras in \(\mathcal{C}\), and under the assumption that \(\underline{1}\) is a left \(\otimes\)-generator, they show that a coalgebra in \(\mathcal{C}\) is (heavily) coseparable if and only if the forgetful functor from the category of right \(C\)-comodules in \(\mathcal{C}\) to \(\mathcal{C}\) is (heavily) separable. If \(A\) is an algebra in \(\mathcal{C}\), let \(\mathcal{T}_A^\#\) be the monoidal category of right transfer morphisms through \(A\). It is discussed when the forgetful functor from the category of entwined modules associated to a cowreath to the base category is heavily separable, and non-trivial examples of heavily separable coalgebras are constructed in \(\mathcal{T}_{A\otimes H^{op}}^\#\), where \(A\) is a Clifford algebra and \(H\) is Sweedler's Hopf algebra.