Kerodon

Corollary 9.1.7.6. Let $\operatorname{\mathcal{C}}$ be a small filtered category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets having colimit $K = \varinjlim (\mathscr {F})$. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and which admits $\mathscr {F}(C)$-indexed colimits, for each $C \in \operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{D}}$ also admits $K$-indexed colimits. Moreover, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a functor which preserves both $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and $\mathscr {F}(C)$-indexed colimits for each $C \in \operatorname{\mathcal{C}}$, then $G$ also preserves $K$-indexed colimits.