Proposition 9.2.9.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every triple of regular cardinals $\kappa \leq \lambda \leq \mu $, the transitivity functor $T: \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ is fully faithful.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 9.2.3.1, it will suffice to show that every object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $(\lambda ,\mu )$-compact when viewed as an object of $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$. This follows from Proposition 9.2.3.1. $\square$