Trying to unify techniques used to compute derived functors I have noticed a pattern when dealing with different types of derived functors.
Let $\mathcal A$ be an abelian category with enough injectives. If $F: \mathcal A \rightarrow \mathcal A$ is a left exact functor we can define its derived functor $R^*F(A)$ by finding an injective resolution of the object $A$, applying $F$, taking cohomology and shifting the complex on step to the left assuming the boundary arrows go from left to right.
We can describe this situation in terms of model categories since $Ch^*(\mathcal A)$ is a model category in the sense of the nlab, theorem $2.2$. I.e, weak equivalences are quasi-isomorphisms and cofibrations are the monomorphisms.
Now $R^*F$ is simply the right derived functor of the induced functor $F':Ch^*(\mathcal A) \rightarrow Ch^*(\mathcal A)$ restricted along $\mathcal A \rightarrow Ch^*(\mathcal A)[\mathcal W^{-1}]$ which sends an object $c$ to the cochain complex consisting of $c$ in degree $0$ and $0$ everywhere else.
Switching gears now. Let $C$ be a right proper model category, we can construct the homotopy limit functor as the right derived functor of the limit functor $\text{lim}: C^I \rightarrow C$ for an appropriate model category structure on $C^I$.
For both $F$ and $\text{lim}$ we essentially constructed the right derived functor valued at $X$ by finding a fibrant replacement $X \rightarrow RX$ and applying the functor to that fibrant replacement. However in both of these cases we see that in practice it suffices to take a weaker form of replacement.
For $F$ we often use the fact that an $F-$acyclic resolution can take the place of an injective resolution.
And for $\text{lim}$ we let $I$ be the category $c \xrightarrow{f} a \xleftarrow{g} b$, then a diagram $\eta: I \rightarrow C$ is fibrant if both $\eta f$ and $\eta g$ are fibrations. But if $C$ is right proper we see that the pullback diagram

is in fact a homotopy pullback if at least one of the morphisms $\eta f$ or $\eta g$ is a fibration. Both need not be fibrations. This is theorem 4.4 on this nlab page.
We see that in both cases, to get the derived functor of a functor $G$ valued at an object $X$ we can take a "weak" fibrant replacement $X \rightarrow (WQ)X$ and apply $G$ to $(WQ)X$ and we will see that $G((WQ)X) \approx G(QX)$ in the homotopy category.
Sorry for the long build up but what is actually going on here category theoretically? Is there some way to unify these two notions of "weak" fibrant replacements in a nice way? I can't for the life of me figure it out.
Thank you in advance!
 A: As mentioned this is examples of computing derived functors using deformations.
A left deformation on category $\mathscr{C}$ with weak equivalences is a functor $Q: \mathscr{C}\to \mathscr{C}$ with a natural weak equivalence $q:Q\to \mathrm{id}_{\mathscr{C}}$. Note this implies that $Q$ preserves weak equivalences.
Now suppose we a given a functor $F:\mathscr{C}\to \mathscr{D}$ between categories with weak equivalences, such that $F$ preserves weak equivalences when restricted to the full subcategory spanned by the essential image of $Q$. In this case we say that $Q$ is a left deformation for $F$.
Now one can show that in this case the left derived functor $\mathbb{L}F$ of $F$ can be computed by the formula
$$
\mathbb{L}F(X)=FQ(X).
$$
Now I have not defined by left derived functors, but i assume this is what you where interested in. I really recommend reading through the 2nd chapter of Emily Riehl's "Categorical homotopy theory" in which she explains this in great detail. She also proves the existence of homotopy colimits (that is a left derived functor for the colimit functor $\mathscr{C}^{I}\to \mathscr{C}$) using the ideas presented in chapter 2.
