Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and Déglise define the notion of a *descent structure* on $\A$, to which corresponds a model structure on the category $C(\A)$ of complexes, where weak equivalences are quasi-isomorphisms. See the nlab page for some background.

The example I am interested in is the case where $\A$ is the category of modules over a ringed space. In example 2.3, they give a descent structure on this category which they show is *flat* (example 3.1), which means that it induces a symmetric monoidal model structure on $C(\A)$ (proposition 3.2). This is interesting because you automatically get a derived tensor product on the homotopy category (i.e. the derived category of $\A$); the usual injective model structure is not monoidal.

At the end of the day I am interested in the derived category $D(\A)$, so I was wondering about the derived functors you get this way ($Lf^*$, $Rf_*$, $\otimes^L$ etc). Are they basically the same as the usual ones, as defined for example in Lipman's book? That is, are cofibrant and fibrant replacements really the same as taking resolutions? Is it possible to get an explicit description for the cofibrant and fibrant objects in this model category?

I'm a beginner at model categories so I hope this question is not too elementary for this site.