Theorem 5.5.2.21 (01ZS)

Theorem 5.5.2.21. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$ with the following property:

$(\ast )$: For every morphism $f: X \rightarrow Y$ and every object $Z \in \operatorname{\mathcal{C}}$, the morphism of simplicial sets $ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$ is a Kan fibration.

Then the coslice comparison morphism $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$ of Construction 5.5.2.17 is an equivalence of $\infty $-categories.

Proof of Theorem 5.5.2.21 from Proposition 5.5.2.22. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category and let $X$ be an object of $\operatorname{\mathcal{C}}$ which satisfies hypothesis $(\ast )$ of Theorem 5.5.2.21. Applying Proposition 5.5.2.22 in the case $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}_{X/}$, we conclude that the coslice comparison morphism $c': \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}}_{X/} ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})_{X/}$ is fully faithful. Since $c'$ is bijective on vertices, it is also essentially surjective, and is therefore an equivalence of $\infty $-categories by virtue of Theorem 4.6.2.20. $\square$