My question is about the construction of derived functor in the language of model categories. (As it is done for example the paper by Dwyer and Spalinski "Homotopy Theories and Model Categories".) I've just started learning these things, so my question may be trivial.
When one looks at the model category of (non-negatively graded) chain complexes, the procedure of taking cofibrant replacement is taking projective resolution of a given complex (Cartan-Eilenberg resolution). This gives a standard recipe for computing derived functor via projective resolution.
However, it is a fact that in order to compute derived functors of an additive left exact functor $F: A \to B$ one can take resolution by any adapted to $F$ class of objects $R$. Actually, there is an equivalence (induced by inclusion) between the derived categories of (non-negatively graded) complexes of objects from $R$ and the derived category of $A$. Class of injective (projective) objects is adapted to any functor. (See Gelfand and Manin III.6 for example.)
How can one prove these facts in the language of model categories?
Thanks for your help,