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Corollary 5.3.5.9. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets. Then the morphism $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ of Construction 5.3.4.11 is a categorical equivalence of simplicial sets.

Proof. Using Proposition 4.1.3.2, we can choose a diagram of $\infty $-categories $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ and a levelwise categorical equivalence $\alpha : \mathscr {F} \rightarrow \mathscr {F}'$. We then have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [r]^{\lambda _{t}} \ar [d] & \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \ar [d] \\ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \ar [r]^{ \lambda '_{t} } & \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}) } \]

where the horizontal maps are given by Construction 5.3.4.11 and the vertical maps are induced by the natural transformation $\alpha $. Since $\alpha $ is a levelwise categorical equivalence, Variant 5.3.2.19 and Corollary 5.3.3.17 guarantee that the vertical maps are categorical equivalences of simplicial sets. Consequently, to show that $\lambda _{t}$ is a categorical equivalence, it will suffice to show that $\lambda '_{t}$ is a categorical equivalence. This is a special case of Theorem 5.3.5.7, since $\lambda '_{t}$ is a scaffold of the cocartesian fibration $\operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (Proposition 5.3.4.17). $\square$