Following Neeman’s article “New axioms for triangulated categories”, for a triangulated category $mathscr T$ let $CT(mathscr T)$ denote the category of *candidate triangles*, i.e. diagrams

begin{equation}Xoverset fto Yoverset gto Zoverset hto Sigma Xquad (*)end{equation}

such that $gf=0$, $hg=0$ and $(Sigma f)h=0$, with morphisms being commutative diagrams between such triangles.

We can define homotopy of maps between candidate triangles to be chain homotopy and there is an automorphism $tildeSigmacolon CT(mathscr T)to CT(mathscr T)$ which takes $(*)$ to

$$Yoverset{-g}to Zoverset{-h}to Sigma Xoverset{-Sigma f}to Sigma Y.$$

We can define mapping cones as in a usual chain complex category, and a lot of the usual results hold for this category (e.g. homotopic maps have isomorphic mapping cones).

What I fail to see, is why the mapping cone construction along with $tildeSigma$ does not give rise to a triangulation of $CT(mathscr T)$?