Kerodon

Proof. Let $\operatorname{\mathcal{K}}$ be a $\lambda $-filtered $\infty $-category, and fix a regular cardinal $\mu \geq \lambda $ such that $\operatorname{\mathcal{K}}$ is $\mu $-small. Applying Proposition 9.4.6.25, we see that the tautological map $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories. Since $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ admits $\operatorname{\mathcal{K}}$-indexed colimits, it follows that $\operatorname{\mathcal{C}}$ admits $\operatorname{\mathcal{K}}$-indexed colimits. We will complete the proof by showing that every functor of $\infty $-categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\operatorname{\mathcal{K}}$-indexed colimits. Enlarging $\operatorname{\mathcal{D}}$ if necessary, we may assume that it admits $\mu $-small $\lambda $-filtered colimits (for example, we can replace $\operatorname{\mathcal{D}}$ by the $\infty $-category $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{D}})$). In this case, $F$ admits an $\operatorname{Ind}_{\lambda }^{\mu }$-extension $\widehat{F}: \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ (Definition 9.3.1.12). Then $\widehat{F}$ preserves $\operatorname{\mathcal{K}}$-indexed colimits and $F$ is isomorphic to the composite functor $\widehat{F} \circ h$. Since $h$ is an equivalence of $\infty $-categories, it follows that $F$ also preserves $\operatorname{\mathcal{K}}$-indexed colimits. $\square$