Corollary 8.3.3.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be an infinite cardinal. Then there exists a fully faithful functor $F: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$, where $\widehat{\operatorname{\mathcal{C}}}$ is $\kappa $-complete and $\kappa $-cocomplete. Moreover, we can arrange that $F$ preserves the limits of all $\kappa $-small diagrams which exist in $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Remark 4.7.3.19, we can choose an uncountable cardinal $\lambda $ of exponential cofinality $\geq \kappa $. Enlarging $\lambda $ if necessary, we may assume that $\operatorname{\mathcal{C}}$ is locally $\lambda $-small. Let $\widehat{\operatorname{\mathcal{C}}}$ denote the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda } )$ and let $F = h_{\bullet }$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{S}}^{< \lambda }$ is $\kappa $-complete and $\kappa $-cocomplete (Remark 7.4.5.7 and Variant 7.4.5.8), the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ has the same property (Remark 7.6.7.5). Moreover, the functor $F$ is fully faithful (Theorem 8.3.3.13) and preserves limits of $\kappa $-small diagrams (Remark 8.3.3.16). $\square$