Lately I've worked on an example that for a group $G$ the $G$-graded vector spaces $G$-vect and $G$-rep, the category of representations on $G$, are Morita equivalent.
I did that by taking the finite dimensional vector spaces vect$_k$ ($k$ a field) as a right module category over $G$-vect. Then I showed that $\mathrm{Fun}_{G-\mathrm{vect}}(\mathrm{vect}, \mathrm{vect})$ is equivalent to $G$-rep and by that those two are Morita equivalent.
It holds that $$k[G]-\mathrm{mod} \cong G-\mathrm{rep}$$ and $$k[G]-\mathrm{comod} \cong G-\mathrm{vect}.$$
One can now ask if a similar Morita equivalence holds for an arbitrary finite dimensional Hopf algebra $H$. It is known that $H-\mathrm{comod} \cong H^*-\mathrm{mod}$. So the question basically is:
Are $H-\mathrm{mod}$ and $H^*-\mathrm{mod}$ Morita equivalent?
As modules over a Hopf algebra are defined as vector spaces with an action (in my case at least) one can again take the vector spaces $\mathrm{vect}$ as the module category. But afterwards in my original proof I used especially representation-properties which I now don't have anymore.
Has someone worked with this before? I'm grateful for anything - tips on how to handle this, literature-tips or maybe even concrete tips on the proof (but I think maybe my question is too wide for this).