Kerodon

\begin{eqnarray} \label{equation:relation-semisimplicial-identity3} \widetilde{\sigma }^{j}_{n} \circ \widetilde{\delta }_{n+1}^{i} & = & \begin{cases} \widetilde{\delta }_{n}^{i} \circ \widetilde{\sigma }^{j-1}_{n-1} & \text{ if } i < j \\ \operatorname{id}_{ [n] } & \text{ if } i = j \text{ or } i = j + 1 \\ \widetilde{\delta }^{i-1}_{n} \circ \widetilde{\sigma }^{j}_{n-1} & \text{ if } i > j+1. \end{cases}\end{eqnarray}