Subfunctor of internal Hom 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.
 A: I am not an expert, but I think that this result is an instance of a more general phaenomenon. If this does not apply to your situation, feel free to ignore my answer or to downvote it.
It looks to me that your framework is a special case of the one described and developed in the following paper.

Hopf monads on monoidal categories, Bruguières, Lack, Virelizier. Advances in Mathematics, 227(2):745-800, 2011.

The motto of the paper is: If T is a good monad on a closed monoidal category, then its category of algebras is also closed and the monadic forgetful functor preserves internal-homs.
A: Doc, let us discern what catechism you are coddling. We know that
$$hom_H(X,Y)=hom_{\mathbb C}(X,Y)^H.$$
Thus, your basic functor does not take values in the category you write. Instead,
$$hom_H(X,-)=mod_H \rightarrow mod_{\mathbb C}.$$
Its left adjoint functor you know and cherish already:
$$-\otimes X= mod_{\mathbb C} \rightarrow mod_{H}.$$
If you impertinently persist with a functor to $mod_H$, you are taking a composition
$$mod_H \xrightarrow{hom_H(X,-)} mod_{\mathbb C}\xrightarrow{T} mod_{H}$$
where the functor $T$ treats a vector space as a trivial $H$-module. The left adjoint to $T$ is the coinvariant functor
$$(-)_H: mod_H \rightarrow mod_{\mathbb C}.$$
Blending these two judicious remarks together, we conclude that the left adjoint to 
$$hom_H(X,-)=mod_H \rightarrow mod_{H}$$
is
$$(-)_H\otimes X: mod_H \rightarrow mod_{H}.$$
Gee, ain’t I a stinker?
