Construction 9.2.9.2. Let $\kappa \leq \lambda \leq \mu $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let us abuse notation by identifying $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with a full subcategory of $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ (Remark 9.2.9.1). We let
\[ T: \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}}) \]
denote the $\operatorname{Ind}_{\lambda }^{\mu }$-extension of the inclusion functor $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ (see Definition 9.2.1.13). We will refer to $T$ as the transitivity functor.