Kerodon

Proof. Fix a regular cardinal $\mu > \lambda $ having exponential cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}$ is $\mu $-small. As in the proof of Corollary 9.3.4.17, we observe that $\operatorname{\mathcal{C}}$ can be realized as the colimit of a $\mu $-small $\lambda $-filtered diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \mu }$, where each $\operatorname{\mathcal{C}}_{\alpha }$ is $\lambda $-small. For each index $\alpha $, let $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ denote a $\lambda $-cocompletion of $\operatorname{\mathcal{C}}_{\alpha }$. It follows from Corollary 9.1.9.11 that $\widehat{\operatorname{\mathcal{C}}}$ can be identified with the colimit of the diagram $\{ \widehat{\operatorname{\mathcal{C}}}_{\alpha } \} $, so every object of $\widehat{\operatorname{\mathcal{C}}}$ can be lifted to an object of $\widehat{\operatorname{\mathcal{C}}}_{\alpha }$ for some index $\alpha $. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}_{\alpha }$ and thereby to reduce to the case where $\operatorname{\mathcal{C}}$ is $\lambda $-small. In this case, we can assume without loss of generality that $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ and that $h$ is the covariant Yoneda embedding of $\operatorname{\mathcal{C}}$ (Theorem 8.4.3.2). The desired result now follows from Corollary 8.4.3.8. $\square$