Let $E^\bullet$ and $F^\bullet$ be complexes on an abelian category; what does it mean to say that $E^\bullet$ and $F^\bullet$ are quasi-isomorphic?
Does it only mean that there is a map of complexes $f:E^\bullet \to F^\bullet$ that induces isomoprhisms between the cohomology objects?
Or does it also guarantee the existence of a map of complexes $g:F^\bullet \to E^\bullet$ inducing the inverses of $H^pf:H^p(E^\bullet)\to H^p(F^\bullet)$?
Put in another way: is quasi-isomorphism an equivalence relation?