Kerodon

Remark 9.3.7.3. In the situation of Proposition 9.3.7.2, we can identify $\widehat{\operatorname{\mathcal{C}}}$ with the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ spanned by the $\kappa $-flat functors (Theorem 9.3.4.14). Consequently, to show that $\widehat{\operatorname{\mathcal{C}}}$ has a final object, it suffices to show that the constant functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \{ \Delta ^0 \} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is $\kappa $-flat. This is a reformulation of the hypothesis that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, since $\mathscr {F}$ is the contravariant transport representation for the right fibration $\operatorname{id}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$.