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:$$\lim_{X' \underset{qis}\to X} Hom^{\bullet}(X',Y) \widetilde\to \lim_{X' \underset{qis}\to X,Y \underset{qis}\to Y'} Hom^{\bullet}(X',Y') \widetilde\leftarrow \lim_{Y \underset{qis}\to Y'}Hom^{\bullet}(X,Y')$$