Functoriality of the principality criterion

Proof. It suffices to check the face and degeneracy maps. For the first face map, we have \begin {align*} \Phi _{n-1}(d_0^*(p,g_1,\dots ,g_n)) &=\Phi _{n-1}(pg_1,g_2,\dots ,g_n) \\ &=(pg_1,pg_1g_2,\dots ,pg_1g_2\dots g_n) =d_0^*\Phi _n(p,g_1,\dots ,g_n). \end {align*}

For a degeneracy map, inserting the identity element among the \(g_i\) inserts a repeated partial product among \(p,pg_1,\dots ,pg_1\dots g_n\), as required. The remaining face maps either multiply two adjacent \(g_i\) or omit the last one, and the same direct calculation proves compatibility. □