Kerodon

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 Corollary 5.3.3.16). 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). $\square$