Proof of Theorem 9.4.3.4.
Let $\kappa \leq \lambda $ be regular cardinals and let $F_{-}: \operatorname{\mathcal{C}}_{-} \rightarrow \operatorname{\mathcal{C}}$ and $F_{+}: \operatorname{\mathcal{C}}_{+} \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories which satisfy conditions $(a)$, $(b)$, and $(c)$ of Theorem 9.4.3.4. Let $\operatorname{\mathcal{C}}_{\pm }$ be the oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ and let $\operatorname{\mathcal{C}}_{\pm }^{0}$ denote the full subcategory of $\operatorname{\mathcal{C}}_{\pm }$ spanned by those objects $(C_{-}, C_{+}, u)$ where $C_{-}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{-}$ and $C_{+}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{+}$. Proposition 7.1.9.4 guarantees that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ is $(\kappa ,\lambda )$-cocomplete. Let $\operatorname{\mathcal{C}}_{\pm }^{1} \subseteq \operatorname{\mathcal{C}}_{\pm }$ be the smallest full subcategory which contains $\operatorname{\mathcal{C}}_{\pm }^{0}$ and is closed under $\lambda $-small $\kappa $-filtered colimits. Note that every object of $\operatorname{\mathcal{C}}_{\pm }^{0}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{\mathcal{C}}_{\pm }$ (Proposition 9.2.8.3), and therefore also as an object of $\operatorname{\mathcal{C}}_{\pm }^{1}$. Applying Proposition 9.3.2.3, we conclude that the inclusion functor $\operatorname{\mathcal{C}}_{\pm }^{0} \hookrightarrow \operatorname{\mathcal{C}}_{\pm }^{1}$ exhibits $\operatorname{\mathcal{C}}_{\pm }^{1}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_{\pm }^{0}$. It follows that the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }^{1}$ is $(\kappa ,\lambda )$-compactly generated, and that an object of $\operatorname{\mathcal{C}}_{\pm }^{1}$ is $(\kappa ,\lambda )$-compact if and only if it a retract of an object of $\operatorname{\mathcal{C}}_{\pm }^{0}$ (Lemma 9.4.1.18). Since $\operatorname{\mathcal{C}}_{\pm }^{0}$ is closed under retracts (Remark 9.2.5.18), every such retract is already contained in $\operatorname{\mathcal{C}}_{\pm }^{0}$. We will complete the proof by showing that $\operatorname{\mathcal{C}}_{\pm }^{1} = \operatorname{\mathcal{C}}_{\pm }$.
Let us say that an object $C_{-} \in \operatorname{\mathcal{C}}_{-}$ is good if, for every object $C_{+} \in \operatorname{\mathcal{C}}_{+}$ and every morphism $u: F_{-}(C_{-} ) \rightarrow F_{+}( C_{+} )$, the triple $( C_{-}, C_{+}, u)$ belongs to the full subcategory $\operatorname{\mathcal{C}}_{\pm }^{1} \subseteq \operatorname{\mathcal{C}}$. We wish to show that every object of $\operatorname{\mathcal{C}}_{-}$ is good. Since $\operatorname{\mathcal{C}}_{-}$ is $(\kappa ,\lambda )$-compactly generated, it will suffice to prove the following:
- $(a)$
The collection of good objects of $\operatorname{\mathcal{C}}_{-}$ is closed under $\lambda $-small $\kappa $-filtered colimits.
- $(b)$
Every $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}_{-}$ is good.
We first prove $(a)$. Suppose we are given a $\lambda $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{J}}$ and a colimit diagram $G_{-}: \operatorname{\mathcal{J}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{-}$ which carries each object of $\operatorname{\mathcal{J}}$ to a good object of $\operatorname{\mathcal{C}}_{-}$; we wish to show that the object $C_{-} = G_{-}(P)$ is also good, where $P$ denotes the cone point of $\operatorname{\mathcal{J}}^{\triangleright }$. Fix a morphism $u: F_{-}( C_{-} ) \rightarrow F_{+}(C_{+} )$ as above. Since the projection map $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \{ C_{+} \} \rightarrow \operatorname{\mathcal{C}}_{-}$ is a right fibration and the inclusion map $\{ P\} \hookrightarrow \operatorname{\mathcal{J}}^{\triangleright }$ is right anodyne (Example 4.3.7.11), we can lift $G_{-}$ to a diagram $G_{\pm }: \operatorname{\mathcal{J}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\pm }$ such that $G_{\pm }( P ) = (C_{-}, C_{+}, u)$ and the composition $\operatorname{\mathcal{J}}^{\triangleright } \xrightarrow {G_{\pm } } \operatorname{\mathcal{C}}_{\pm } \rightarrow \operatorname{\mathcal{C}}_{+}$ is the constant functor taking the value $C_{+}$. Applying Lemma 7.1.9.3, we see that $G_{\pm }$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$. By construction, the restriction $G_{\pm }|_{ \operatorname{\mathcal{J}}}$ factors through $\operatorname{\mathcal{C}}_{\pm }^{1}$, so that $G_{\pm }(P) = (C_{-}, C_{+}, u)$ also belongs to $\operatorname{\mathcal{C}}_{\pm }^{1}$.
We now prove $(b)$. For the remainder of the proof, we fix an object $C_{-} \in \operatorname{\mathcal{C}}_{-}$ which is $(\kappa ,\lambda )$-compact; we wish to show that $C_{-}$ is good. Let us say that an object $C_{+} \in \operatorname{\mathcal{C}}_{+}$ is good if, for every morphism $u: F_{-}( C_{-} ) \rightarrow F_{+}( C_{+} )$ in the $\infty $-category $\operatorname{\mathcal{C}}$, the triple $C_{\pm } = ( C_{-}, C_{+}, u)$ is contained in $\operatorname{\mathcal{C}}_{\pm }^{1}$. Note that, if $C_{+}$ is $(\kappa ,\lambda )$-compact, then this condition is automatic (since $C_{\pm }$ is an object of the subcategory $\operatorname{\mathcal{C}}_{\pm }^{0} \subseteq \operatorname{\mathcal{C}}_{\pm }^{1}$). Using our assumption that $\operatorname{\mathcal{C}}_{+}$ is $(\kappa ,\lambda )$-compactly generated, we are reduced to proving the following:
- $(c)$
The collection of good objects of $\operatorname{\mathcal{C}}_{+}$ is closed under $\lambda $-small $\kappa $-filtered colimits.
Fix a $\lambda $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$ and a colimit diagram $H: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{+}$ carrying each object of $\operatorname{\mathcal{K}}$ to a good object of $\operatorname{\mathcal{C}}_{+}$; we wish to show that $C_{+} = H(Q)$ is also good, where $Q \in \operatorname{\mathcal{K}}^{\triangleright }$ denotes the cone point. Set $\operatorname{\mathcal{E}}= \{ C_{-} \} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}$ and $\overline{\operatorname{\mathcal{E}}} = \{ C_{-} \} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}^{\triangleright }$, so that projection onto the second factors determines left fibrations
\[ U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{K}}\quad \quad \overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{K}}^{\triangleright }. \]
Fix an uncountable cardinal $\mu $ of cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small, so that the left fibration $\overline{U}$ admits a covariant transport representation given by the composition
\[ \operatorname{\mathcal{K}}^{\triangleright } \xrightarrow {H} \operatorname{\mathcal{C}}_{+} \xrightarrow {F_{+}} \operatorname{\mathcal{C}}\xrightarrow { \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F_{-}(C_{-}), \bullet ) } \operatorname{\mathcal{S}}_{< \mu } \]
Since the functor $F_{+}$ is $(\kappa ,\lambda )$-finitely and the object $F_{-}(C_{-} )$ is $(\kappa ,\lambda )$-compact, this composite functor is a colimit diagram in $\operatorname{\mathcal{S}}_{< \mu }$. Applying Corollary 7.4.3.14, we see that the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \overline{\operatorname{\mathcal{E}}}$ is a weak homotopy equivalence.
Fix a morphism $u: F_{-}(C_{-} ) \rightarrow F_{+}(C_{+} )$ in the $\infty $-category $\operatorname{\mathcal{C}}$; we wish to show that the triple $C_{\pm } = (C_{-}, C_{+}, u)$ belongs to the full subcategory $\operatorname{\mathcal{C}}_{\pm }^{1} \subseteq \operatorname{\mathcal{C}}_{\pm }$. Let us regard the triple $(C_{-}, Q, u)$ as an object $\widetilde{Q} \in \overline{\operatorname{\mathcal{E}}}$ satisfying $\overline{U}( \widetilde{Q} ) = Q$. Since $\overline{U}$ is a left fibration, the induced map
\[ \overline{V}: \overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} } \rightarrow \overline{\operatorname{\mathcal{E}}} \times _{ \operatorname{\mathcal{K}}^{\triangleright } } ( \operatorname{\mathcal{K}}^{\triangleright } )_{ / Q } \simeq \overline{\operatorname{\mathcal{E}}} \]
is a Kan fibration (Proposition 4.3.7.2). Set $\operatorname{\mathcal{E}}_{ / \widetilde{Q} } = \operatorname{\mathcal{E}}\times _{ \overline{\operatorname{\mathcal{E}}} } \overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} }$ so that projection onto the second factor induces Kan fibration $V: \operatorname{\mathcal{E}}_{ / \widetilde{Q} } \rightarrow \operatorname{\mathcal{E}}$. Note that the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} }$ has a final object (Proposition 4.7.3.20) and is therefore weakly contractible (Corollary 4.7.3.23). It follows from Corollary 3.3.7.4 that the inclusion map $\operatorname{\mathcal{E}}_{ / \widetilde{Q} } \hookrightarrow \overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} }$ is also a weak homotopy equivalence, so that the $\infty $-category $\operatorname{\mathcal{E}}_{ / \widetilde{Q} }$ is weakly contractible. Applying Lemma 9.4.3.15 to the left fibration $(U \circ V): \operatorname{\mathcal{E}}_{ / \widetilde{Q} } \rightarrow \operatorname{\mathcal{K}}$, we can choose a weakly contractible $\lambda $-small subset $\operatorname{\mathcal{K}}' \subseteq \operatorname{\mathcal{E}}_{ / \widetilde{Q} }$ for which the restriction $(U \circ V)|_{ \operatorname{\mathcal{K}}' }: \operatorname{\mathcal{K}}' \rightarrow \operatorname{\mathcal{K}}$ is a left fibration. Applying Corollary 9.1.5.14 (and Remark 9.1.3.7), we conclude that $\operatorname{\mathcal{K}}'$ is a $\kappa $-filtered $\infty $-category and that the functor $T = (U \circ V)|_{ \operatorname{\mathcal{K}}' }$ is right cofinal. The composition
\[ \operatorname{\mathcal{K}}'^{\triangleright } \hookrightarrow \overline{\operatorname{\mathcal{E}}}_{ / \widetilde{Q} }^{\triangleright } \rightarrow \overline{\operatorname{\mathcal{E}}} = \{ C_{-} \} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+} \]
determines a diagram $H': \operatorname{\mathcal{K}}'^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{\pm }$, whose image in $\operatorname{\mathcal{C}}_{-}$ is the constant diagram taking the value $C_{-}$ and whose image in $\operatorname{\mathcal{C}}_{+}$ is given by the composition $H \circ T^{\triangleright }$. Applying Corollary 7.2.2.3 and Lemma 7.1.9.3, we deduce that $H'$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_{\pm }$ Our assumptionon $H$ guarantees that $H'|_{ \operatorname{\mathcal{K}}'}$ factors through the full subcategory $\operatorname{\mathcal{C}}^{1}_{\pm }$. Since $\operatorname{\mathcal{C}}^{1}_{\pm }$ is closed under $\lambda $-small $\kappa $-filtered colimits, we conclude that the object $C_{\pm } = (C_{-}, C_{+}, u)$ also belong to $\operatorname{\mathcal{C}}^{1}_{\pm }$.
$\square$