\begin{tikzcd}
 & & & \sigma_1\sigma_0 \arrow[d] & & & \\
 & & & (\sigma_1, \sigma_0) & & & \\
 & & \sigma_1\sigma_2 \arrow[dr] & & & & \\
 & \sigma_1 \arrow[rr] \arrow[d] \arrow[uurr, bend left] & & (\sigma_1, \sigma_2) & & \sigma_2 \arrow[ll] \arrow[d] \arrow[dll] & \\
\sigma_1\sigma \arrow[r] & (\sigma_1, \sigma) \arrow[urr] & & (\sigma, \sigma_2) \arrow[u] \arrow[dr] & & (\sigma_2, \sigma_0) \arrow[dl] & \sigma_0\sigma_2 \arrow[l] \\
 & & \sigma \arrow[ul] \arrow[ur] \arrow[rr] & & (\sigma, \sigma_0) & & \sigma_0 \arrow[ul] \arrow[ll] \arrow[uuuulll, bend right] \\
 & & & \sigma\sigma_2 \arrow[uu, dashed] & \sigma\sigma_0 \arrow[u] & &
\end{tikzcd}