For $G,H$ Hopf algebras, and $\pi:G \to H$ a Hopf algebra map, can some-one give me an example of a (right) $H$-comodule $(V,\Delta_R)$, such that $$ (G \otimes V)^{\text{co}H} = \lbrace g_{(1)} \otimes v_{(0)} \otimes \pi(g_{(2)})v_{(1)} = g \otimes v \otimes 1 ~ | ~ g \otimes v \in G \otimes V) \rbrace = \lbrace 0 \rbrace. $$

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Edit: Ideally, I'd like $G$ to be the coordinate algebra of a Drinfeld--Jimbo quantum group. But I'm interested in other (non-trivial) examples also.