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Remark 9.2.1.20. 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 9.2.1.19 (for any sufficiently large $\mu $), we obtain a functor $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, which is characterized (up to isomorphism) by the requirements that it is $(\kappa ,\lambda )$-finitary and that the diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [r] \ar [d]^{F} & \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{Ind}_{\kappa }^{\lambda }(F) } \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}}) } \]

commutes up to isomorphism. In other words, $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ can be identified with the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ (see Definition 9.2.1.13).