Q:

Can one extend a morphism of commutative triangles to a morphism of  octahedral diagrams?

Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular faces'). Is it true than one can extend to a morphism of such diagrams:
1. any morphism of one of the 'commutative faces' of the octahedron
2 any morphism of the pair of morphisms whose target is the upper vertex of the octahedron (i.e. a morphism of commutative triangles not lying on the faces of the octahedrons)?
Is there any text where I could look for various facts of this sort?
P.S. It seems that the answer is 'no' in general. Having a morphism of 'commutative faces', one can extend it to a morphism of three neighbouring 'triangulated faces'. Thus one obtains morphisms of each of six vertices. Yet (all possible) compositions of edges of the 'first' commutative triangles and the neigbouring distinguished faces do not yield all edges of the octahedron; two of the edges (in the 'lower hat') are missing.
Yet it would be very interesting to know which additional conditions are needed in order for the morphism of the octahedrons desired to exist. I would be deeply grateful for any comments!!

A:

It is true in a Heller triangulated category aka $\infty$-triangulated category (although strictly speaking one only needs a 3-triangulation for octahedra) that any morphism between the bases of octahedra (by which I mean the 3 objects and two composable morphisms from which the octahedron is built) extends to a morphism between the octahedra (it is part of the axiomatics).
It turns out that the triangulated categories which turn up in "nature" all satisfy these stronger axioms. For instance the homotopy category of a stable model category is $\infty$-triangulated (a more general statement holds which is Theorem 2 in Maltsiniotis' preprint).
References are M. Künzer. Heller triangulated categories and G. Maltsiniotis. Catégories Triangulées Supérieures

A:

The identity of commutative triangles does not lift to a morphism of Verdier octahedra in general, for it may happen that there exist two mutually nonisomorphic Verdier octahedra on the same commutative triangle.
Cf. http://www.math.rwth-aachen.de/~kuenzer/counterexample.pdf . This has been observed already by Amnon Neeman in the 90s (unpublished). 
(The situation is better if one uses 3-triangles, i.e. "distinguished octahedra", in the sense of Heller/oo-triangulated categories. Cf. http://www.math.rwth-aachen.de/~kuenzer/heller.pdf , Lem. 3.2. So there exists a choice of octahedra such that morphisms of commutative triangles lift to morphisms of these particular octahedra.)

