Octahedral axiom

Let $\mathcal T$ be a pretriangulated category, and let $X\xrightarrow{f}Y\xrightarrow{g}Z$ be composable morphisms. Complete $f$, $g$, and $gf$ to distinguished triangles, with third objects $C_f$, $C_g$, and $C_{gf}$. The octahedral axiom asserts that these triangles can be connected by morphisms

so that the standard octahedral diagram commutes and

is a distinguished triangle.