Kerodon

Remark 9.2.9.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\kappa \leq \lambda $ be regular cardinals, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ (Variant 9.2.1.7). Suppose we are given regular cardinals $\kappa '$ and $\lambda '$ satisfying $\kappa \leq \kappa ' \leq \lambda ' \leq \lambda $, and let $\widehat{\operatorname{\mathcal{C}}}_0$ be the smallest full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ which contains the image of $h$ and is closed under $\lambda '$-small $\kappa '$-filtered colimits. It follows from Proposition 9.2.3.3 that the functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}_0$ exhibits $\widehat{\operatorname{\mathcal{C}}}_0$ as an $\operatorname{Ind}_{\kappa '}^{\lambda '}$-completion of $\operatorname{\mathcal{C}}$. Stated more informally, we can identify $\operatorname{Ind}_{\kappa '}^{\lambda '}(\operatorname{\mathcal{C}})$ with the full subcategory of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ generated by $\operatorname{\mathcal{C}}$ under $\lambda '$-small $\kappa '$-filtered colimits.