Suppose we have a functor between two model categories $F\colon \mathcal{C}\to \mathcal{D}$. Suppose we don't know that it is a part of a Quillen pair. Nevertheless, it can still happen that the derived functor $\mathbb{L}F\colon \mathrm{Ho}(\mathcal{C})\to \mathrm{Ho}(\mathcal{D})$ exists. Then we apriori don't know that it can be computed by the formula $$\mathbb{L}F(A)=\gamma F(R)\phantom{aaaa}(*)$$ for some cofibrant replacement $R\to A$ of $A$ (here $\gamma$ denotes the localization funtor).

Can we somehow extend (or maybe we should say "complete") the categories $\mathcal{C}$ and $\mathcal{D}$ to $\hat{\mathcal{C}}$ and $\hat{\mathcal{D}}$, and extend the functor $F$ to $\hat{F}$ so that there will exist $\mathbb{L}\hat{F}$ which will restrict to $\mathbb{L}F$, and such that $\hat{F}$ can be computed by the formula $(*)$?

It feels like there should exist something analogous to the case of abelian categories. There, if we don't have enough resolutions, we can always complete our categories and extend the functor in the desired way (like extending category of coherent sheaves to quasi-coherent sheaves in order to have injective resolutions).

Or maybe we can extend $\mathcal{C},\mathcal{D},F$ in such a way that $\hat{F}$ becomes a part of a Quillen pair? Is there any result like that?

Thank you very much for your help!

minimalsuch extensions. Maybe one can embed the model categories $\mathcal{C}$ and $\mathcal{D}$ into something much bigger... – Sasha Patotski Aug 12 '14 at 21:24