For an abelian category $\mathcal{A}$, define the homotopy category of chain complexes $\mathcal{K}(\mathcal{A})=\mathcal{C}(\mathcal{A})/\mathcal{I},$ where $\mathcal{C}(\mathcal{A})$ denotes the category of chain complexes over $A$ and $\mathcal{I}$ denotes the ideal of null-homotopic maps. Denote $\mathcal{W}$ the set of all homotopy equivalences in $\mathcal{C}(\mathcal{A})$
What I would like to know is the following:
Is $\mathcal{K}(\mathcal{A})$ canonicaly isomorphic to $\mathcal{C}(\mathcal{A})[\mathcal{W}^{-1}]$? That is, does the quotient functor $\mathcal{C}(\mathcal{A}) \rightarrow \mathcal{K}(\mathcal{A})$ satisfy the universal property of localization with respect to $\mathcal{W}$?
I am fairly certain that it is true if the localization is additive functor between additive categories. I know this is true provided that $\mathcal{W}$ is a right/left multiplicative system (i.e. satisfy a version of Ore conditions), however, I don't know whether this is the case.
Therefore, a sub-question is
Is $\mathcal{W}$ in general a right/left multiplication system?
(I should also mention that I am a little bit suspicious about this: If the claim is true, then, since $\mathcal{W}$ is a subset of quasi-isomorphisms, this would imply that the canonical functor to the derived category is just the localization with respect to quasi-isomorphisms and the step of passing to the homotopy category is there for some copmutational convenience only.)
Thanks in advance for any help.