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Let $H$ be a Hopf algebra, and let ${\cal M}^H_H$ the category of right $H$-module-comodules, that is, the objects of ${\cal M}^H_H$ are right $H$-modules, and right $H$-comodules, such that, for $V$ an object in ${\cal M}^H_H$, we have $$ (vh)_{(0)} \otimes (vh)_{(1)} = v_{(0)}h_{(1)} \otimes v_{(1)}h_{(2)}, \text{ for all } v \in V, h \in H. $$ My question is what monodical structures can we put on ${\cal M}^H_H$? For $V,W \in {\cal M}^H_H$, defining $V \otimes W$ to be the ordinary tensor product of comodules, and them defining the right $H$-action to be multiplication on $W$ alone works. Whereas the same approach with the $H$-action defined according to $$ (v \otimes w) m = vm_{(1)} \otimes vm_{(2)} $$ does not work. Can anyone think of other general examples?

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Notice that that category is equivalent to that of vector spaces. You can use this equivalence to transport the tensor product on the latter. –  Mariano Suárez-Alvarez Aug 3 '12 at 23:27
Yes I see this. The question is how should I define a right $H$-action on this tensor product? –  Ago S Aug 3 '12 at 23:38

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