Kerodon

Corollary 7.4.5.5. 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 $\operatorname{\mathcal{C}}$. Let $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be the cocartesian fibration of Definition 5.3.3.1, and let $W$ be the collection of $U$-cocartesian morphisms of $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$. Then the localization $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})[W^{-1}]$ is a colimit of the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\mathscr {F}): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{QC}}$.