Proposition 9.3.7.2. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-filtered and essentially $\lambda $-small, and let $F: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. Then the colimit $X = \varinjlim (F)$ is final object of $\widehat{\operatorname{\mathcal{C}}}$. In particular, the $\infty $-category $\widehat{\operatorname{\mathcal{C}}}$ has a final object.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We wish to show that, for every object $Y \in \widehat{\operatorname{\mathcal{C}}}$, the morphism space $\operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(Y, X)$ is contractible. Using Proposition 7.4.1.22, we see that the collection of objects $Y \in \widehat{\operatorname{\mathcal{C}}}$ which satisfy this condition is closed under colimits; in particular, it is closed under $\lambda $-small $\kappa $-filtered colimits. We may therefore assume without loss of generality that $Y = F(C)$ for some object $C \in \operatorname{\mathcal{C}}$. Fix a regular cardinal $\mu \geq \lambda $ such that $\widehat{\operatorname{\mathcal{C}}}$ is locally $\mu $-small, so that $F(C)$ corepresents a functor
\[ G: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}_{< \mu } \quad \quad Z \mapsto \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}( F(C), Z ). \]
We wish to show that $G(X)$ is contractible. Since $F(C)$ is a $(\kappa ,\lambda )$-compact object of $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 9.3.2.3), we can identify $G(X)$ with the colimit of the diagram $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$. Since $F$ is fully faithful, we can identify $G \circ F$ with the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet )$, so that $\varinjlim (G \circ F)$ is contractible by virtue of Example 7.4.3.7. $\square$