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.
Best Answer
The uniqueness condition of the maps between cones is very restrictive. If it holds for every commutative square, this indeed means that you could define a "cone functor" $\mathrm{Mor}(\mathcal T) \to \mathcal T$ from the category of morphisms of your triangulated category $\mathcal T$ to $\mathcal T$ itself (just choose a cone object for each morphism, then your uniqueness condition ensures that the cone functor is well defined on morphisms). It turns out that this makes $\mathcal T$ a semisimple abelian category, if $\mathcal T$ is assumed to be Karoubian (i.e. every idempotent splits; many common triangulated categories are Karoubian). I found a proof of this claim in the following article:
http://www.math.uni-bielefeld.de/~gstevens/no_functorial_cones.pdf
In conclusion: you can't expect your uniqueness condition to hold globally in "useful" triangulated categories. There is a technique to overcome this problem, that is, using pre-triangulated dg-categories (introduced by Bondal and Kapranov) to "lift" triangulated categories. In this new framework you indeed have functorial cones.
Perhaps this doesn't answer your specific question (which, as I understand, is about a given commutative square), but it should point out that the desired uniqueness is, roughly speaking, very difficult to obtain in general.