Hovey introduces the notion of a closed monoidal structure and a closed monoidal functor. Then he goes on to say that this naturally gives rise to the notion of closed modules over a closed monoidal category. I am a bit confused about what this means and how this definition 'interacts' with the definition of modules over monoidal model categories.
in particular I am not quite sure what additional structure a closed module has. So let $\mathcal{C}$ be a closed monoidal category. Let $\mathcal{D}$ be a closed $\mathcal{C}$-module. That means there is a functor
$$ \otimes: \mathcal{D}\times \mathcal{C}\to\mathcal{D} $$
and associativity and unit transformations, subject to the standard axioms.
Now a monoidal model category is automatically closed and if $\mathcal{M}$ is such a monoidal model category a $\mathcal{M}$-model category is by definition required to be a model category $\mathcal{N}$ making
$$ \otimes :\mathcal{N}\times\mathcal{M}\to\mathcal{N} $$ into a Quillen bifunctor. Is it then automatically a closed module, when forgetting the model structure?
