Proposition 9.2.3.1. Let $\kappa \leq \lambda $ be regular cardinals and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories satisfying the following conditions:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ admits $\lambda $-small, $\kappa $-filtered colimits.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
For each object $C \in \operatorname{\mathcal{C}}$, the image $f(C)$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{D}}$.
Then the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $f$ is a fully faithful functor $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$.