I am currently reading some notes on homological algebra, more specifically derived categories. After introducing (distinguished) triangles of complexes the author states the following theorem:
Now this is a little bit confusing to me as to what he means by an exact sequenec in $D(\mathcal{A})$. I would assume that $u$, $v$ and $w$ are morphisms in the category of chain complexes, because otherwise I don't know how the homology morphisms are given. When he says exact in $D(\mathcal{A})$, does he mean distinguished?
The proof of the theorem is then given:
As I don't understand what the assumptions of the theorem are I have trouble understanding why the first sequence in the proof is quasi isomorphic to (III.19).