TITLE: left derived functors commute with filtered  colimits
QUESTION [7 upvotes]: Let $\mathcal{A}$ be an $\mathbf{AB5}$ category with enough projectives and let $F:\mathcal{A}\rightarrow\mathcal{B}$ be a right exact functor into abelian category that commutes with filtered colimits. What reasonable assumptions should one imposed on $\mathcal{A}$ and $F$ to obtain that letf derived functors of $F$ commute with filtered colimits?

REPLY [2 votes]: This is only a comment on some aspect of the Brandenburg answere. 
I wish elucidate the aspect about the $I$-naturality of projective resolutions:
considerind a example of a single morphism $f: A\to B$ and two projective resolution $P_\ast\to A$ and  $Q_\ast\to B$ then is a well knowed fundamental lemma of homological algebrathat exist (homotopically unique) a extentions of $f$ to the (augmented) chain complexes.
In the basic homological algebra text there is also the example of a resolution of exat sequences. 
But what if insted a single morphism (or a exat sequence) we consider a small diagram?
From  Weibel, "An introduction to homological algebra", 2.3.13 on p.43 (I see it from link text) follow that $\mathcal{A}^I$ has enought projectives, but isnt clear if for a projective $P\in \mathcal{A}^I$ each $P(i)\in \mathcal{A}$ is projective as we need in the context of the above Brandenburg proof 
(this is true if for any $i\in I$  the right Kan extention of the $i$-valutation $v(i): \mathcal{A}^I\to \mathcal{A}$ is exact, I dont know if this follow from the "$I$ is filtrant" hypothesis).
From the book  "Theory of Categories (BArry Mitchell)  cor.7.6 p. 138,   let  $T_i: \mathcal{A}^I\to \mathcal{A}$ ($i\in I$) the $i$-valuation, and $S_i$ its left adjuction (the left Kan extention), now for a projective $P\in \mathcal{A}$ the object $S_i(P)(j),\ j\in I$ is a sum of copies of $P$ (see the Weibel reference above) then is projective. THe above  corollary assert that  projectives of $\mathcal{A}^I$ are objects of the form $\bigoplus_{i\in I}S_i(P_i)$ (where $P_i\in \mathcal{A}$ is a prjective) and all its retracts. 
This is enought for ensure the existence a projective resolution of a  $(X_i)_i\in \mathcal{A}^I$ with projective arguments.