In this paper, we study modules over quotient spaces of certain categorified fiber bundles. These are understood as modules over entwining structures involving a small K-linear category $${\cal D}$$ and a K-coalgebra C. We obtain Frobenius and separability conditions for functors on entwined modules. We also introduce the notion of a C-Galois extension $${\cal E} \subseteq {\cal D}$$ of categories. Under suitable conditions, we show that entwined modules over a C-Galois extension may be described as modules over the subcategory $${\cal E}$$ of C-coinvariants of $${\cal D}$$ .