# Kerodon

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Corollary 7.4.3.12. Let $\operatorname{\mathcal{C}}$ be a small simplicial set, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram, let $U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map, and let $W$ be the collection of all $U$-cocartesian morphisms of $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$. Then the localization $( \int _{\operatorname{\mathcal{C}}} \mathscr {F})[W^{-1} ]$ is a colimit of the diagram $\mathscr {F}$ in the $\infty$-category $\operatorname{\mathcal{QC}}$.

Proof. Apply Corollary 7.4.3.11 to the cocartesian fibration $\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}$. $\square$