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.



Trivial Hopf Coinvariant Subspace Example
Zero comodule!!


If you want something more halloweeny, take $G$ to be a field, $H$ - a Hopf algebra, not a field, $V$ - any simple nontrivial $H$-comodule... 


**EDIT:** as asked, $G$ is any Hopf algebra with a simple non-trivial comodoule, $H=G\otimes G$, the map is $x\mapsto x \otimes 1$, $V$ is a simple nontrivial $1\otimes G$-comodule (and hence $H$-comodule)... Bingo!


