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Proposition 5.4.8.6 (Functoriality). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of simplicial categories. Assume that:
For every pair of objects $C,C' \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,C')_{\bullet }$ is an $\infty $-category.
For every pair of objects $D,D' \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{D}}}(D,D')_{\bullet }$ is an $\infty $-category.
Then the induced map $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F): \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$ is a functor of $(\infty ,2)$-categories: that is, it carries thin $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ to thin $2$-simplices of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$.
Proof.
It follows from Theorem 5.4.8.1 that the simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$ are $(\infty ,2)$-categories. We will show that the morphism $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is a functor by verifying the criterion of Proposition 5.4.7.9. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms in the category $\operatorname{\mathcal{C}}$ (or, equivalently, in the $(\infty ,2)$-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$). Then $f$ and $g$ determine a $2$-simplex of the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we identify with $2$-simplex $\sigma $ of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (see Remark 2.4.3.8). By virtue of Example 5.4.8.3, $\sigma $ is a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and its image $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)(\sigma )$ is a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$.
$\square$