We show how the metric, the almost complex structure and the almost product structure of the homogeneous nearly Kähler $\mathbb{S}^3\times\mathbb{S}^3$ can be recovered from a submersion $\pi:\mathbb{S}^3\times\mathbb{S}^3\times\mathbb{S}^3\to\mathbb{S}^3\times\mathbb{S}^3$. On $\mathbb{S}^3\times\mathbb{S}^3\times\mathbb{S}^3$ we have the maps obtained either by changing two coordinates, or by cyclic permutations.