Let $\mathcal{H}$ be a Hopf algebra over $\mathbb{C}$. Let $\textrm{mod}_\mathcal{H}$ be the monoidal abelian category of finite-dimensional modules over $\mathcal{H}$. Fix $X\in\textrm{Obj}(\textrm{mod}_\mathcal{H})$. We know that the functor $$(-\otimes X):\textrm{mod}_\mathcal{H}\rightarrow\textrm{mod}_\mathcal{H}$$ is left adjoint to functor $$\textrm{Hom}_{\mathbb{C}}(X,-):\textrm{mod}_\mathcal{H}\rightarrow\textrm{mod}_\mathcal{H}.$$

Moreover, $\textrm{Hom}_{\mathcal{H}}(X,Y)$ is a submodule of $\textrm{Hom}_{\mathbb{C}}(X,Y)$, for all $Y\in\textrm{Obj}(\textrm{mod}_\mathcal{H})$. It seems that $\textrm{Hom}_{\mathcal{H}}(X,-)$ defines a subfunctor of $\textrm{Hom}_{\mathbb{C}}(X,-)$, since composition of morphisms of modules is a morphism of modules, as well. Am I right about that? If this is the case, does $\textrm{Hom}_{\mathcal{H}}(X,-)$ have any (left or right) adjoints? Does anyone know of any references that deal with this? Thanks in advance for answers.