On the cohomology of comodules over smash coproducts

We consider the category of comodules over a smash coproduct coalgebra. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from colinear morphisms over the smash coproduction, colinear morphisms over het underlying algebra and rational colinear morphisms over the underlying ecoalgebra. We give several applications and connect our results to existing spectral sequences in the literature.