  Let $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf
module category $\mathcal C$ as modules over the smash extension $\mathcal C\#
H$. We construct Grothendieck spectral sequences for the cohomologies as well
as the $H$-locally finite cohomologies of these objects. We also introduce
relative $(\mathcal D,H)$-Hopf modules over a Hopf comodule category $\mathcal
D$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra
$A$. We construct Grothendieck spectral sequences for their cohomologies by
using their rational $Hom$ objects and higher derived functors of coinvariants.
