While trying to understand the model structure in a category of dg-modules over a dga, I found the following excerpt from Toën's [The homotopy theory of dg-categories and derived Morita theory]:
... the above definition endows $C-Mod$ with a structure of a cofibrantly generated model category (see for example [Hi, 11]). The natural $C(k)$-enrichment of $C-Mod$ endows furthermore $C-Mod$ with a structure of a $C(k)$-model category in the sense of [Ho1, 4.2.18]...
This is where he defines a model structure in $C-Mod = C(k)^C$, when $C$ is a dg category.
I looked up [Hi, 11], which was constructing a cofibrantly generated model structure on $M^C$, where $M$ has a cofibrantly generated model structure and $C$ is a category.
In order to apply this to our setting, where the domain category is also a dg-category, we need some enriched version of the [Hi, 11] I suppose, but I am confused how this can be done.
I am trying to work out the case of $C$ having a single object (so $Hom(\cdot,\cdot)=A$ is a dga), but then what would be the generating cofibrations: by unraveling [Hi, 11], it seems to be something like $\left((0 \to k \to 0) \otimes A\right) \longrightarrow \left((0 \to k \to k \to 0) \otimes A\right)$, but this doesn't seem to agree with another characterization usually given in terms of $P$-resolutions as in Deriving DG categories.
How can I understand the above excerpt? Or there is a somewhere else that I can read about cofibrately generated model category structure on $C-Mod$?
