\Sigma X'
        }$$ whose rows are triangles, there exists a morphism $w:Z\longrightarrow Z'$ such that the resulting diagram also commutes (that is, $wg=g'v$ and $h'w = (\Sigma u)h$).

(T4)

:   (Octahedral axiom.) Assume that $$X \stackrel{f}{\longrightarrow} Y \stackrel{h}{\longrightarrow} Z' \stackrel{i}{\longrightarrow} \Sigma X, \quad 
        Y \stackrel{g}{\longrightarrow} Z \stackrel{j}{\longrightarrow} X' \stackrel{k}{\longrightarrow} \Sigma Y, \quad
        X \stackrel{gf}{\longrightarrow} Z \stackrel{\ell}{\longrightarrow} Y' \stackrel{m}{\longrightarrow} \Sigma X$$ are triangles, and arrange them as in the following picture

    $\phantom{xxxx}$

    0;&lt;1pt,0pt&gt;:&lt;0pt,-1pt&gt;:: (105,0) \*+[Y’]{} =“0”, (74,90) \*+[X]{} =“1”, (207,90) \*+[Z]{} =“2”, (0,69) \*+[Z’]{} =“3”, (103,147) \*+[Y]{} =“4”, (133,69) \*+[X’]{} =“5”, “0”, [|+\^m “1”]{}, “2”, [\_“0”]{}, “3”, [\^[p]{} “0”]{}, “0”, [@[.&gt;]{}\^[q]{} “5”]{}, “1”, [\^[gf]{} “2”]{}, “3”, [|+\_i “1”]{}, “1”, [\^f “4”]{}, “4”, [\^g “2”]{}, “2”, [@[.&gt;]{}\_j “5”]{}, “4”, [\^h “3”]{}, “5”, [@[.&gt;]{}|+\_[(h)k]{}“3”]{}, “5”, [@[.&gt;]{}\^k |+“4”]{},

    $\!\!\!\!\!\!\!$where a “$+$” on an arrow $A\to B$ means a morphism $A\to \Sigma B$. Then there exist morphisms $p:Z'\to Y'$ and $q:Y'\to X'$ such that $$Z' \stackrel{p}{\longrightarrow} Y' \stackrel{q}{\longrightarrow} X' \stackrel{(\Sigma h)k}{\longrightarrow} \Sigma Z'$$ is a triangle, and we have $ph = \ell g, (\Sigma f)m = kq, i=mp$ and $j=q\ell$.

    In other words, the four oriented triangles in the above pictures are triangles of ${\mathcal{T}}$, the four non-oriented triangles are commutative diagrams, and the two 