Kerodon

Example 9.2.1.9. Following the convention of Remark 4.7.0.5, a cardinal $\kappa $ is small if it satisfies $\kappa < \operatorname{\textnormal{\cjRL {t}}}$, where $\operatorname{\textnormal{\cjRL {t}}}$ is some fixed strongly inaccessible cardinal. In this case, a functor of $\infty $-categories $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.2.1.4) if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\operatorname{\textnormal{\cjRL {t}}}}$-completion of $\operatorname{\mathcal{C}}$ (in the sense of Variant 9.2.1.7). In particular, $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$ if and only if it exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\aleph _0}^{\operatorname{\textnormal{\cjRL {t}}}}$-completion of $\operatorname{\mathcal{C}}$. Stated more informally, we have equivalences