Lemma 9.5.5.17. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-cocomplete, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor of $\infty $-categories. Then $h$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ if and only if it satisfies conditions $(0)$, $(1)$, and $(2)$ of Lemma 9.5.5.16, together with the following additional condition:
- $(3)$
The objects $\{ h(C) \} _{C \in \operatorname{\mathcal{C}}}$ generate $\widehat{\operatorname{\mathcal{C}}}$ under $\lambda $-small colimits. That is, if a full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ contains the essential image of $h$ and is closed under $\lambda $-small colimits, then $\widehat{\operatorname{\mathcal{C}}}' = \widehat{\operatorname{\mathcal{C}}}$.