This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?. I am deeply grateful for the contributions there; they roughly say that given a morphism of commutative triangles in a triangulated category one:
1) cannot extend it (in general) to a morphism of certain fixed octahedra "constructed from" these triangles.
2) is not able to prove the existence of such a morphism for some possible choice of octahedra of this sort (for general triangulated categories, without assuming certain additional axioms).
Now my question is: which "parts" of the morphism of octahedra diagram can be chosen to be commutative (in two settings: if the octahedra are fixed or not)? In particular, for $X\to Y\to Z$ and $X'\to Y'\to Z'$ I would like to consider the cubic diagram whose vertices are $X,Y,Cone(Y\to Z)[-1],Cone(X\to Z)[-1]$ + the corresponding vertices for the remaining triangle (i.e., $X',Y',Cone(Y'\to Z')[-1],Cone('X\to Z')[-1]$). Can one "make it commutative"?
Any hints and references would be very welcome!
