I have for any $n$ some distinguished triangles $$X_n->Y_n->Z_n$$ $$X'_n->Y'_n->Z'_n$$ The families $(X_n), (Y_n), (X'_n), (Y'_n)$ are inductive systems, $(X_n)$ is quasi-isomorphic to $(X'_n)$, $(Y_n)$ is quasi-isomorphic to $(Y'_n)$ and these quasi-isomorhisms are compatible with the maps $(u_n):(X_n)->(Y_n)$ and $(v_n):(X'_n)->(Y'_n)$.

Can I deduce a quasi-isomorphism of inductive systems between $(Z_n)$ and $(Z'_n)$? If not any other conditions on $(u_n)$ and $(v_n)$ that would allow to get the same conclusion?

homotopicmaps between the cones. @Theo: it sounds like you're claiming this is actually false. Is there a good way to see that? $\endgroup$ – Anton Geraschenko Feb 7 '11 at 18:07