Given a Hopf algebra $H$, I wonder when the monoidal category of $H$-modules is left-closed, right-closed, and finally, under what circumstances right and left internal hom are isomorphic.
If I'm not mistaken, it seems that for $H$-modules $M,N$ one always has a natural $H$-action on $\text{Hom}(M,N)$ given by $(h.f)(m) := h^{\prime} f(S(h^{\prime\prime})m)$, and that provided with this $H$-action, evaluation and coevaluation $\text{Hom}(M,N)\otimes M\to N$ and $N\to\text{Hom}(M,N\otimes M)$ are $H$-linear. In particular $\text{Hom}(M,-)$ provides a right adjoint to $-\otimes M$ in $H\text{-Mod}$.
If the antipode $S$ of $H$ is invertible, one can put another $H$-action on $\text{Hom}(M,N)$, namely $(h.f)(m) := h^{\prime\prime} f(S^{-1}(h^{\prime})m)$, and this yields a right adjoint to $M\otimes -$ in $H\text{-Mod}$.
1) Where can I read about these things? I'm sure it is written down somewhere, but wasn't able to find a reference.
2) Under what circumstances are the two $H$-actions on $\text{Hom}(M,N)$ equal, or at least isomorphic? For example, they are equal if $H$ is cocommutative (in which case in particular $S=S^{-1}$). Is commutativity also sufficient?
Thank you, Hanno