Kerodon

Proof. Let $\mu \geq \lambda $ be an uncountable cardinal of exponential cofinality $\geq \kappa $ such that $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is locally $\mu $-small, and let $h^{\bullet }: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu } )$ be a contravariant Yoneda embedding for $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. By virtue of Proposition 7.4.1.18, it will suffice to show that the composite functor $(h^{\bullet } \circ F^{\operatorname{op}}): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu } )$ preserves $\kappa $-small limits.

Since $F$ carries each object of $\operatorname{\mathcal{C}}$ to a $(\kappa ,\lambda )$-compact object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ (Proposition 9.2.3.3), the functor $h^{\bullet } \circ F^{\operatorname{op}}$ factors through the full subcategory $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu })$ of $(\kappa ,\lambda )$-finitary functors from $\operatorname{Ind}(\operatorname{\mathcal{C}})$ to $\operatorname{\mathcal{S}}_{< \mu }$. Since $\lambda $-small $\kappa $-filtered colimits commute with $\kappa $-small limits in $\operatorname{\mathcal{S}}_{< \mu }$ (Proposition 9.1.5.8), the subcategory $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu })$ is closed under $\kappa $-small limits. It will therefore suffice to show that $(h^{\bullet } \circ F^{\operatorname{op}}): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu })$ preserves finite limits. Since $F$ exhibits $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $F^{\ast }: \operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. We are therefore reduced to showing that the composite functor

preserves $\kappa $-small limits. Since $F$ is fully faithful, this composition is a contravariant Yoneda embedding for the $\infty $-category $\operatorname{\mathcal{C}}$, and therefore preserves all limits which exist in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (Proposition 7.4.1.18). $\square$