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Proposition 9.3.7.11. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\lambda $-small $\kappa $-filtered $\infty $-categories and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $F$ is right cofinal.
- $(2)$
The functor $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ is right cofinal.
- $(3)$
The functor $\operatorname{Ind}_{\kappa }^{\lambda }(F)$ preserves final objects.
Proof.
Since $\operatorname{\mathcal{C}}$ is $\lambda $-small and $\kappa $-filtered, the $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ has a final object (Proposition 9.3.7.2). The equivalence $(2) \Leftrightarrow (3)$ now follows from Corollary 7.2.1.9, and the implication $(2) \Rightarrow (1)$ is a special case of Lemma 9.3.7.10. We complete the proof by showing that $(1) \Rightarrow (3)$. Assume that $F$ is right cofinal and let $X$ be a final object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$; we wish to show that $\widehat{F} = \operatorname{Ind}_{\kappa }^{\lambda }(F)$ carries $X$ to a final object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$. Using Proposition 9.3.7.2, we can identify $X$ with a colimit of the tautological map $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Since the functor $\widehat{F}$ is $(\kappa ,\lambda )$-finitary, it follows that $\widehat{F}(X)$ is a colimit of the diagram $\widehat{F} \circ h$, which is isomorphic to the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$. If $F$ is right cofinal, then this is also a colimit of the tautological map $\operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ (Corollary 7.2.2.3), and is therefore a final object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ (Proposition 9.3.7.2).
$\square$