Notation 8.1.1.6 (Projection Maps). Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is equipped with projection maps
\[ \lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \quad \quad \lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}. \]
Here $\lambda _{+}$ carries an $n$-simplex $\sigma $ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to the $n$-simplex of $\operatorname{\mathcal{C}}$ given by the composition
\[ \Delta ^ n = \operatorname{N}_{\bullet }( [n] ) \hookrightarrow \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow {\sigma } \operatorname{\mathcal{C}}, \]
while $\lambda _{-}$ carries $\sigma $ to the $n$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$ given by the composite map
\[ (\Delta ^ n)^{\operatorname{op}} = \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} ) \hookrightarrow \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow {\sigma } \operatorname{\mathcal{C}}. \]
Concretely, $\lambda _{-}$ and $\lambda _{+}$ are given on vertices by the formulae $\lambda _{-}( f: X \rightarrow Y ) = X$ and $\lambda _{+}( f: X \rightarrow Y ) = Y$.