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Proposition 7.5.8.1. 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 $\operatorname{\mathcal{C}}$, and let $W$ denote the collection of horizontal edges of the homotopy colimit $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ (Definition 5.3.4.1). Then an $\infty $-category $\operatorname{\mathcal{D}}$ is a colimit of the diagram

\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]

if and only if it is a localization of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ with respect to $W$, in the sense of Remark 6.3.2.2.

Proof. Let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the projection map of Definition 5.3.3.1 and let $W'$ denote the collection of all $U$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. Choose a functor of $\infty $-categories $T: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ with respect to $W'$. Let $\lambda _{t}: \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the taut scaffold of Construction 5.3.4.11. Then $\lambda _{t}$ is a categorical equivalence of simplicial sets (Corollary 5.3.5.9). Moreover, a morphism of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ belongs to $W'$ if and only if it is isomorphic (as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$) to an element of $\lambda _{t}(W)$ (see Proposition 5.3.3.15). It follows that the composite map $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F}) \xrightarrow { \lambda _{t} } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \xrightarrow {T} \operatorname{\mathcal{D}}$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $ \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} )$ with respect to $W$. We conclude by observing that $\operatorname{\mathcal{D}}$ is a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F})$ (Corollary 7.4.3.16). $\square$