Defining hom spaces in the derived category as limits of hom spaces in the homotopy category

Question

Let $C$ be an abelian category and $K(C)$ the homotopy category of complexes in $C$. I've seen the following claimed in several sources (without proof):

A. The following isomorphisms hold: $$\lim_{X' \underset{qis}\to X} Hom_{K(C)}(X',Y) \widetilde\to \lim_{X' \underset{qis}\to X,Y \underset{qis}\to Y'} Hom_{K(C)}(X',Y') \widetilde\leftarrow \lim_{Y \underset{qis}\to Y'}Hom_{K(C)}(X,Y') $$

The limits (actually colimits in this case) are taken over all quasi-isomorphisms.

Why is this true? And how can one prove this?

Can this statement be upgraded to a statement about the internal Hom bifunctor? Something along the lines of:

B. The following quasi-isomorphisms hold: $$\lim_{X' \underset{qis}\to X} Hom^{\bullet}(X',Y) \widetilde\to \lim_{X' \underset{qis}\to X,Y \underset{qis}\to Y'} Hom^{\bullet}(X',Y') \widetilde\leftarrow \lim_{Y \underset{qis}\to Y'}Hom^{\bullet}(X,Y')$$

Answer 1

This is actually a standard property of the homotopy category of complexes on which construction of the derived category is based, formulated in unusual way.

For a quasi-isomorphism $s:X'\to X$ and $f\in Hom(X',Y)$ denote by $fs^{-1}$ the image of $f$ in the first colimit(in a second we will justify the choice of notation). By description of the filtered colimit, two images $fs^{-1}$ and $gt^{-1}$(for $t:X''\to X,g\in Hom(X'',Y)$) coincide iff there exists $X'''$ and qisms $p:X'''\to X',q:X'''\to X''$ and a map $r\in Hom(X''',Y)$ such that $sp=tq,fp=gq$. This is exactly the condition for fractions $fs^{-1}$ and $gt^{-1}$ to be equivalent in the right localization by qisms(see e.g. Weibel, 10.3).

Analogously, the third colimit can be identified with the set of equivalence classes of left fractions. Now, Ore conditions guarantee that these two sets of equivalence classes coincide, which is precisely the isomorphism you are looking for.

Edit: As Denis-Charles Cisinski explains, the answer to B is negative. Indeed, let's pick a non-zero object $A$ in $C$ and consider, for instance, $X=A[0], Y=A\xrightarrow{1} A$ where $Y$ is a contractible complex concentrated in degrees $-1,0$. The second and third colimits are zero as the category of quasi-isomorphisms $Y\to Y'$ has a final object $Y\to 0$. But the first colimit is not zero: for $X'=X$ there is a non-zero element $Id_A\in Hom(A,A)=Hom^0(X,Y)$ which survives in the colimit because for a quasi-isomorphism $X'\to X$ the map $(X')^0\to A$ has to be non-zero.