Kerodon

Proof. Without loss of generality, we may assume that $\lambda $ is uncountable (see Example 9.2.1.10). 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 9.2.1.23 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, we can use Theorem 9.2.4.14 to identify $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}})$ with the $\infty $-category $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, and the desired result follows from Lemma 9.2.4.17. $\square$