simplicial identity of commuting face maps

Given the simplex category, for \(i \leq j\), it holds for the face map

\begin{align*} \delta_i \circ \delta_j = \delta_{j-1} \circ \delta_{i} \end{align*}

\(\delta_i \circ \delta_j\) first inserts a zero at the \(j\)-th spot, then another at the \(i\)-th spot. Thus the zero at the \(j\)-th spot gets shifted by 1 to the \(j+1\)-th spot.

\(\delta_{j+1} \circ \delta_{i}\) inserts a zero at the \(i\)-th spot, then another at the \(j+1\)-th spot