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I have some trouble regarding the answer to this question. My problem with it has been mentioned in the comments below it, and I think adressed in an answer, but I can't understand this second answer.

So I'd actually like to know what the definite answer is. The context is the following : we have $\mathbf{AB5}$ abelian categories $A,B$, and a right exact functor $F:A\to B$. Assume $A$ has enough projectives, so that we can make sense of the left derived functors of $F$. We wish to know when those left-derived functors preserve the filtered (or directed) colimits $I\to A$ that $F$ preserves. An obvious necessary condition is that colimits of projectives in $A$ are $F$-acyclic.

Now if $A$ is concrete enough and really nice, we can make "projective resolution" into a functor $A\to Ch^+(A)$; but in the general case, the "fundamental lemma" that is mentioned in the answer only gives a well-defined functor $A\to K^+(A)$. If there is a "concrete" projective resolution, then I understand the answer with no problem, but I have trouble seeing how to get from there to the desired result, because there does not seem to always be a functorial projective resolution with values in $Ch^+(A)$.

It seems to me if we have a directed system $(C_i)_{i\in I}$, take projective resolutions $(P_{\bullet, i})_{i\in I}$, compute for each $i\leq j$ the (unique up to homotopy) induced chain map $P_{\bullet, i}\to P_{\bullet, j}$, and try to lift this system of chain maps to a functor $I\to Ch^+(A)$, we run into some "cohomological obstruction" (cohomology of $I$ perhaps, it would be interesting if it were actually related to some (co)homology of $I$)

Indeed, what seems to happen is that, denoting $f_{ji}$ a (fixed, concrete) chain map $P_{\bullet, i }\to P_{\bullet, j}$ extending $C_i\to C_j$ whenever $i\leq j$, we get equations like $f_{kj}f_{ji} - f_{ki} = dh_{ijk}+h_{ijk}d$ for $i\leq j \leq k$ and homotopies $h_{ijk}$ and we're looking for $g_{ij}$'s such that $g_{ij}-f_{ij} = d h'_{ij} + h'_{ij}d$ and $g_{kj}g_{ji} - g_{ki} = 0$.

I can't seem to make this intuition precise, but this reminds me a lot of group cohomology, and it looks like there could be some cohomology of $I$ lurking behind all this (but I can't make this precise, because the $f_{ij}$'s don't seem to be a cocyle looking to be a coboundary, as their "differential" -probably something like $f_{kj}f_{ji}-f_{ki}$ - is not $0$, but a coboundary)

So this digression makes me ask a few (related) questions in the end :

(1) How do we resolve the problem I raised in the mentioned answer/how do you prove that under the given conditions, left derived functors preserve filtered (directed if it's easier) colimits ?

(2) Is the given answer actually valid, or does it depend on some more conditions on $I$ ?

(3) If the answer to (2) is that it depends on $I$, can the conditions on $I$ be made "cohomological", in a sense that makes precise the vague intuition I gave at the end of my question ? How ?

(4) If the answer to (2) is that it depends on $I$, and the answer to (3) is that it does depend on some sort of cohomology of $I$, are there more studies on how "cohomology" of indexing categories $I$ influences functor categories $C^I$, for $C$ abelian, or more generally ?

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