Remark 8.1.1.7 (Duality). Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then there is a canonical isomorphism of simplicial sets $\iota : \operatorname{Tw}(\operatorname{\mathcal{C}}) \simeq \operatorname{Tw}(\operatorname{\mathcal{C}}^{\operatorname{op}} )$, given on $n$-simplices by precomposition with the unique isomorphism
\[ \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] )^{\operatorname{op}} \simeq \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ). \]
The isomorphism $\iota $ interchanges the projection maps $\lambda _{-}$ and $\lambda _{+}$ of Notation 8.1.1.6.