Theorem 9.2.4.14. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small. Then the convariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ factors through $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{<\lambda })$, and exhibits $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{<\lambda })$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ (in the sense of Variant 9.2.1.7).
Proof of Theorem 9.2.4.14. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small. We wish to show that the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ exhibits $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. By virtue of Proposition 8.4.5.8, it will suffice to show that $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ is the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ which contains all representable functors and is closed under $\lambda $-small $\kappa $-filtered colimits. This follows from Example 9.2.4.11, Lemma 9.2.4.16, and Lemma 9.2.4.17. $\square$