Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 9.3.5.28. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete and essentially small. Then the covariant Yoneda embedding

\[ h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\kappa -\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \]

exhibits $\operatorname{Fun}^{\kappa -\operatorname{lex}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}$.

Proof. Apply Corollary 9.3.5.27 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal. $\square$