Proposition 8.6.7.2. Let $\operatorname{\mathcal{E}}$ be a locally Kan simplicial category. Then:
- $(1)$
The simplicial category $\operatorname{\mathcal{E}}^{\asymp }$ of Notation 8.6.7.1 is locally Kan.
- $(2)$
The forgetful functor $\pi : \operatorname{\mathcal{E}}^{\asymp } \rightarrow \operatorname{\mathcal{E}}$ is a weak equivalence of simplicial categories (see Definition 4.6.8.7).
- $(3)$
The functor $\pi $ induces an equivalence of $\infty $-categories $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{E}}^{\asymp } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$.