It is well known that non-uniqueness of a cone for a morphism in a triangulated category $C$ makes constructing exact functors (of triangulated categories) a difficult task. In section 3 of his "Some new axioms for triangulated categories" Neeman proposes an alternative axiomatics of triangulated categories (that includes a certain notion of 'good morphism' of distinguished triangles). I wonder whether Neeman's methods can be applied to the motivic stable homotopy category $SH$ of Morel and Voevodsky? Can any better methods for 'rigidifying' (for example, higher categories?) be applied to $SH$?

Upd. It seems that I need an category $G$ whose objects and morphism are morally distinguished triangles in $C$ and their morphisms + some extra data. Any $C$-morphism can be extended to an object of $G$, any commutative square could be extended to a $G$-morphism, whereas for any $G$-morphism of distinguished triangles $X\to Y\stackrel{v}{\to} Z\stackrel{w}{\to} X[1]$ and $X'\to Y'\stackrel{v}{\to} Z'\stackrel{w'}{\to} X'[1]$ that is $0$ on $X$ and $Y$ the corresponinding morphism from $Z$ to $Z'$ equals $v'\circ \theta \circ w$ for some $\theta\in C(X[1],Y')$ (this is the correction of the Neeman's axiom GTR2).

Denis-Charles Cisinski has written some very interesting comments on the relations between various types of 'nice' triangulated categories; yet I wonder which texts treat this topic.