  We develop the categorical algebra of the noncommutative base change of a
comodule category by means of a Grothendieck category $\mathfrak S$. We
describe when the resulting category of comodules is locally finitely
generated, locally noetherian or may be recovered as a coreflective subcategory
of the noncommutative base change of a module category. We also introduce the
category ${_A}\mathfrak S^H$ of relative $(A,H)$-Hopf modules in $\mathfrak S$,
where $H$ is a Hopf algebra and $A$ is a right $H$-comodule algebra. We study
the cohomological theory in ${_A}\mathfrak S^H$ by means of spectral sequences.
Using coinduction functors and functors of coinvariants, we study torsion
theories and how they relate to injective resolutions in ${_A}\mathfrak S^H$.
Finally, we use the theory of associated primes and support in noncommutative
base change of module categories to give direct sum decompositions of minimal
injective resolutions in ${_A}\mathfrak S^H$.
