Corollary 7.3.5.3. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $\operatorname{\mathcal{D}}$ be an $\infty $-category, and let $F_0: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \rightarrow \operatorname{\mathcal{D}}$ be a diagram. The following conditions are equivalent:
- $(1)$
The diagram $F_0$ admits a left Kan extension along the projection map $U: \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
- $(2)$
For every object $C \in \operatorname{\mathcal{C}}$, the diagram
\[ \mathscr {G}(C) \simeq \{ C\} \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \hookrightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \xrightarrow {F_0} \operatorname{\mathcal{D}} \]admits a colimit in the $\infty $-category $\operatorname{\mathcal{D}}$.