Kerodon

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

Corollary 9.2.7.15. Let $\kappa $ be a small regular cardinal. Then the inclusion functor $\operatorname{\mathcal{S}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{S}}$ exhibits $\operatorname{\mathcal{S}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{S}}_{< \kappa }$.

Proof. Apply Corollary 9.2.7.13 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal (here the inequality $\kappa < \lambda $ guarantees that $\kappa \triangleleft \lambda $; see Example 9.1.7.11). $\square$