Corollary 9.3.4.17. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then every object of $\operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{C}})$ can be realized as the colimit of a $\lambda $-small $\kappa $-filtered diagram $\operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$.
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Proof. Without loss of generality, we may assume that $\lambda $ is uncountable (see Example 9.3.1.10). Fix a regular cardinal $\mu > \lambda $ having exponential cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}$ is $\mu $-small. Then $\lambda \triangleleft \mu $ (Proposition 9.1.7.8), so $\operatorname{\mathcal{C}}$ can be realized as the colimit of a $\mu $-small, $\lambda $-filtered diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \mu }$ (see Variant 9.1.7.21), where each $\operatorname{\mathcal{C}}_{\alpha }$ is $\lambda $-small. It follows from Proposition 9.3.3.12 that every object of $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}})$ can be lifted to an object of $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}}_{\alpha } )$, for some index $\alpha $. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}_{\alpha }$, and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is $\lambda $-small. In this case, the desired result follows from Corollary 9.3.4.15. $\square$