  We consider the category of comodules over a smash coproduct coalgebra
$C\smashco H$. We show that there is a Grothendieck spectral sequence
connecting the derived functors of the Hom functors coming from $C\smashco
H$-colinear, $H$-colinear and rational $C$-colinear morphisms. We give several
applications and connect our results to existing spectral sequences in the
literature.
