Kerodon

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Proposition 7.5.7.1. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a (strictly commutative) diagram of $\infty $-categories indexed by $\operatorname{\mathcal{C}}$. Then a Kan complex $X$ is a colimit of the functor $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ if and only if it is weakly homotopy equivalent to the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F})$.

Proof. Let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ be the taut scaffold of Construction 5.3.4.11. Then $\lambda _{t}$ is a categorical equivalence of simplicial sets (Corollary 5.3.5.9), and therefore a weak homotopy equivalence (Remark 4.5.3.4). The desired result now follows from Corollary 7.4.5.7. $\square$