Remark 9.2.1.15. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits.
- $(2)$
The tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ admits a left adjoint $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$.
Moreover, if these conditions are satisfied, then the functor $F$ is the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$. See Proposition 8.4.5.13.