Remark 8.1.1.2. For every integer $n \geq 0$, there is a unique isomorphism of simplicial sets $\operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \simeq \Delta ^{2n+1}$. It follows that, for every simplicial set $\operatorname{\mathcal{C}}$, we can identify $n$-simplices $\sigma $ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ with $(2n+1)$-simplices $\overline{\sigma }$ of $\operatorname{\mathcal{C}}$. In terms of these identifications, the face and degeneracy operators of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ are given explicitly by the formulae
\[ \overline{d^{n}_ i \sigma } = d^{2n}_{n-i} d^{2n+1}_{n+1+i} \overline{\sigma } \quad \quad \overline{s^{n}_ i \sigma } = s^{2n+2}_{n-i} s^{2n+1}_{n+1+i} \overline{\sigma }. \]