Proposition 9.2.3.3. Let $\kappa \leq \lambda $ be regular cardinals and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ if and only if it satisfies conditions $(0)$, $(1)$, and $(2)$ of Proposition 9.2.3.1, together with the following additional condition:
- $(3)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated under $\lambda $-small, $\kappa $-filtered colimits by the image of $f$. That is, if $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ is a full subcategory which contains $f(\operatorname{\mathcal{C}})$ and is closed under $\lambda $-small, $\kappa $-filtered colimits, then $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}$.