Galois measurings for noncommutative base change of entwined contramodule and entwined comodule categories

We study the noncommutative base change of an entwining structure $(A,C,\psi )$ by a Grothendieck category $\mathfrak{S}$, using two module like categories. These are the categories of entwined comodule objects and entwined contramodule objects in $\mathfrak{S}$ over the entwining structure $(A,C,\psi )$. We consider criteria for maps between these noncommutative spaces, induced by generalized maps between entwining structures, known as measurings, to behave like Galois extensions. We also study conditions for extensions of these noncommutative spaces, understood as functors between module like categories, to have separability, Frobenius or Maschke type properties.