Remark 9.3.3.9. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Applying the functor $\operatorname{Ind}_{\kappa }^{\lambda }(-)$ of Notation 9.3.3.8 (for any sufficiently large $\mu $), we obtain a $(\kappa ,\lambda )$-finitary functor $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, for which the diagram
commutes up to isomorphism. It follows that $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ agrees (up to isomorphism) with the functor defined in Notation 9.3.1.13, given by the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$.