Notation 8.6.7.1. Let $\operatorname{\mathcal{E}}$ be a simplicial category. We define a new simplicial category $\operatorname{\mathcal{E}}^{\asymp }$ as follows:
The objects of $\operatorname{\mathcal{E}}^{\asymp }$ are the objects of $\operatorname{\mathcal{E}}$.
For every pair of objects $X,Y \in \operatorname{\mathcal{E}}$, the simplicial set $\operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( X, Y)_{\bullet }$ is the twisted arrow construction $\operatorname{Tw}( \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{\bullet } )$.
For every triple of objects $X,Y,Z \in \operatorname{\mathcal{E}}$, the composition law
\[ \circ : \operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( Y,Z)_{\bullet } \times \operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( X,Y )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{E}}^{\asymp } }( X, Z)_{\bullet } \]is obtained by applying the twisted arrow functor $\operatorname{Tw}$ to the composition law for the simplicial category $\operatorname{\mathcal{E}}$.
The simplicial category $\operatorname{\mathcal{E}}^{\asymp }$ is equipped with a simplicial functor $\pi : \operatorname{\mathcal{E}}^{\asymp } \rightarrow \operatorname{\mathcal{E}}$, which carries each object to itself and is given on morphism spaces by the projection map
described in Notation 8.1.1.6.