If I have two exact triangles $X \to Y \to Z \to X[1]$ and X' \to Y' \to Z' \to X'[1] in a triangulated category, and I have morphisms X \to X', Y \to Y' which 'commute' (i.e., such that $X \to Y \to Y' = X \to X' \to Y'$), thene there exists a (not necessarily unique) map Z \to Z' which completes what we've got to a morphism of triangles.
Is there a criterion which ensures the uniqueness of this cone-map?
I'd like something along the lines of: if \operatorname{Ext}^{-1}(X,Y')=0 then yes.
(I might be too optimistic, cfr. Prop 10.1.17 of Kashiwara-Schapira Categories and Sheaves: in addition to \operatorname{Hom}{(X[1],Y')} = 0 they also assume \operatorname{Hom} {(Y,X')} =0. I really don't have this second assumption.)
(In the case I'm interested in X=X', Y=Y' and X\to X', Y \to Y' are the identity maps.)
(If it makes things easier, although I doubt it, you can take the category to be the bounded derived category of coherent sheaves on some, fairly nasty, scheme.)
In the context I have in mind X, Y, X', Y' are all objects of the heart of a bounded t-structure. If we assumed \operatorname{Hom}{(Z,Y')} = 0 or \operatorname{Hom}{(X[1],Z')} = 0 then the result easily follows. I don't think I'm happy making those assumptions though.