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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).