Kerodon

Corollary 7.5.8.13. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{Set_{\Delta }}$ be a categorical colimit diagram carrying the final object of $\operatorname{\mathcal{C}}^{\triangleright }$ to a simplicial set $K$. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and $\overline{\mathscr {F}}(C)$-indexed colimits, for each object $C \in \operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{D}}$ also admits $K$-indexed colimits. Moreover, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a functor of $\infty $-categories which preserves $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$-indexed colimits and $\overline{\mathscr {F}}(C)$-indexed colimits for each $C \in \operatorname{\mathcal{C}}$, then $G$ also preserves $K$-indexed colimits.