I am defining $D(A)$ by the localization of Ch(A) for $A$ an abelian category at the set of quasi-isomorphisms, and $Q : Ch(A) \to D(A)$ is the localization map.
(So $D(A)$ for this question means: the objects are the same as Ch(A), but we have introduced for each quasi-isomorphism and "edge" between the domain and codomain of that quasi-isomorphism. Morphisms are paths of edges that are subject to certain equivalence relationships, namely: if there are two adjacent edges that are honest morphisms in $Ch(A)$, we can compose them and replace the length 2 piece of that path by their composition. If the fake inverse to a quasi-isomoprhism $s$ is placed next to $s$, we can replace them by the identity morphism.
$Q$ then sends objects to objects, and on morphisms it sends $f : X \to Y$ to the path $f$.)
This construction does not first construct the homotopy category $K(A)$, so this question is not a complete tautology. (It is the relationship between these two constructions of the derived category that I am trying to understand.)
Suppose that $f : X \to X$ is homotopic to the identity. Why is $Q(f) = 1_X$ in the derived category?
I ask because the proof that $Q(h) = Q(g)$ if $h$ and $g$ are homotopic in Gelfand-Manin (page 159, chapter 3 part 4) seems to reduce everything to this case, and I don't see why it is true.
It suffices to show that a nullhomotopic map is sent to zero in the derived category, since the localization functor is additive (I think). But I am at a loss for how to describe chain homotopies in the derived category.
Thank you for your help.