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Theorem 4.6.6.9. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X$ and $Y$, and suppose that the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Then the left-pinched morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(X,Y)$ is also an $\infty $-category, and the comparison map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \]

of Construction 4.6.6.3 is an equivalence of $\infty $-categories.

Proof of Theorem 4.6.6.9. Let $\operatorname{\mathcal{C}}$ be a simplicial category containing a pair of objects $X,Y \in \operatorname{\mathcal{C}}$ for which the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Applying Corollary 4.6.6.25, we deduce that the left-pinched morphism space $\operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X, Y)$ is also an $\infty $-category. We wish to show that the comparison map

\[ \theta : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y) \]

of Construction 4.6.6.3 is an equivalence of $\infty $-categories. To prove this, it will suffice to show that for every simplicial set $K$, postcomposition with $\theta $ induces a bijection

\[ \theta _{K}: \pi _0( \operatorname{Fun}( K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}(K, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y))^{\simeq } ). \]

By virtue of Corollary 4.6.6.26, we can identify $\pi _0( \operatorname{Fun}(K, \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Y))^{\simeq } )$ with the set $\pi _0( \operatorname{Fun}( \Phi (K), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } )$. Under this identification, $\theta _{K}$ corresponds to the map

\[ \pi _0( \operatorname{Fun}( K, \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \Phi (K), \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } )^{\simeq } ) \]

given by precomposition with the map $\rho _{K}: \Phi (K) \rightarrow K$ of Construction 4.6.6.22, which is bijective by virtue of the fact that $\rho _{K}$ is a categorical equivalence of simplicial sets (Proposition 4.6.6.23). $\square$