Two different Hopf module structures on a hom space

Question

Let $H$ be a Hopf algebra over a field $k$ (feel free to assume finite-dimensional and/or semisimple, if that helps), and let $V$ and $W$ be finite dimensional left $H$-modules. As vector spaces, we have isomorphisms $$ V^* \otimes_k W \cong \text{Hom}_k(V,W) \cong W \otimes_k V^*. $$ Chasing through the isomorphisms, this allows us to define two different (left) actions of $H$ on $\text{Hom}_k(V,W)$. The action from the first isomorphism is $$ (h \cdot f)(v) = h_2 f (S(h_1) v), $$ while the action from the second isomorphism is $$ (h \cdot f)(v) = h_1 f (S(h_2) v) $$ (in the above, I'm using sumless Sweedler notation). Is there any particular reason to favour one of these actions over the other?

Answer 1

I think the reason could be the following. The module structure on $\mathsf{Hom}_k(V,W)$ given by $(h\cdot f)(v) = h_1f(S(h_2)v)$ is the one that ensures that $\mathsf{Hom}_k(V,-)$ is the right adjoint of the functor $-\otimes V$ in the category of left $H$-modules. That is to say, that makes of $\mathsf{Hom}_k(V,W)$ the internal hom of the category of left $H$-modules. In particular, if $V$ is finite-dimensional then we know that a right adjoint to $-\otimes V$ is provided by $-\otimes V^*$ and this is in accordance with the fact that the isomorphism $\mathsf{Hom}_k(V,W) \cong W\otimes V^*$ is $H$-linear.