Kerodon

Proof. As in the proof of Proposition 9.2.4.22, we can identify $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ which contains all representable functors and is closed under $\lambda $-small $\kappa $-filtered colimits. It follows from Lemma 9.2.4.16 (and Example 9.2.4.11), we see that every functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ which belongs to $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\kappa $-flat. To complete the proof, it will suffice to show that every $\kappa $-flat functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ satisfying condition $(\ast ')$ also satisfies condition $(\ast )$ of Proposition 9.2.4.22.

Using Proposition 8.6.8.3, we may assume that $\mathscr {F}$ is the contravariant transport representation of an essentially $\lambda $-small right fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Replacing $U$ by a minimal fibration if necessary (Proposition 5.3.8.11). Similarly, we may assume that the $\infty $-category $\operatorname{\mathcal{C}}$ is minimal (Proposition 4.7.6.15). By assumption, the functor $\mathscr {F}$ is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ which is essentially $\lambda $-small. Since $\operatorname{\mathcal{C}}_0$ is also minimal, it is $\lambda $-small. We extend $\operatorname{\mathcal{C}}_0$ to a transfinite sequence of $\lambda $-small full subcategories $\{ \operatorname{\mathcal{C}}_{\alpha } \subseteq \operatorname{\mathcal{C}}\} _{0 \leq \beta \leq \kappa }$ as follows:

• Suppose that $\beta = \alpha +1$ is a successor ordinal, and that the full subcategory $\operatorname{\mathcal{C}}_{\alpha }$ has been defined. Let $\operatorname{\mathcal{E}}_{\alpha } \subseteq \operatorname{\mathcal{E}}$ be the inverse image of $\operatorname{\mathcal{C}}_{\alpha }$. Since $\operatorname{\mathcal{C}}_{\alpha }$ is $\lambda $-small and $U$ is an essentially $\lambda $-small minimal fibration, the $\infty $-category $\operatorname{\mathcal{E}}_{\alpha }$ is also $\lambda $-small. Our assumption that $\mathscr {F}$ is $\kappa $-flat guarantees that the $\infty $-category $\operatorname{\mathcal{E}}$ is $\kappa $-filtered. Since $\kappa \triangleleft \lambda $, Proposition 9.1.7.14 guarantees that $\operatorname{\mathcal{E}}_{\alpha }$ is contained in a $\lambda $-small simplicial subset $\operatorname{\mathcal{E}}^{+}_{\alpha } \subseteq \operatorname{\mathcal{E}}$ which is also a $\kappa $-filtered $\infty $-category. We define $\operatorname{\mathcal{C}}_{\beta }$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the objects $U(X)$, where $X$ belongs to $\operatorname{\mathcal{E}}^{+}_{\alpha }$.

• If $\beta $ is a nonzero limit ordinal, we define $\operatorname{\mathcal{C}}_{\beta }$ to be the union $\bigcup _{\alpha < \beta } \operatorname{\mathcal{C}}_{\alpha }$.

We now claim that the full subcategory $\operatorname{\mathcal{C}}_{\kappa } \subseteq \operatorname{\mathcal{C}}$ satisfies condition $(\ast )$ of Proposition 9.2.4.22. By construction, it is a $\lambda $-small subcategory of $\operatorname{\mathcal{C}}$ which contains $\operatorname{\mathcal{C}}_0$, so that $\mathscr {F}$ is left Kan extended from $\operatorname{\mathcal{C}}_{\kappa }^{\operatorname{op}}$ (Proposition 7.3.8.6). It will therefore suffice to show that the functor $\mathscr {F}|_{ \operatorname{\mathcal{C}}_{\kappa }^{\operatorname{op}} }$ is $\kappa $-flat: that is, that the $\infty $-category $\operatorname{\mathcal{E}}_{\kappa } = \operatorname{\mathcal{C}}_{\kappa } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\kappa $-filtered. This is a special case of Corollary 9.1.5.13, since $\operatorname{\mathcal{E}}_{\kappa }$ can be realized as the colimit of the $\kappa $-filtered diagram $\{ \operatorname{\mathcal{E}}_{\alpha }^{+} \} _{\alpha < \kappa }$ comprised of $\kappa $-filtered $\infty $-categories. $\square$