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.



Subfunctor of internal Hom
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.



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


