Proposition 8.3.7.6. The diagram of $\infty $-categories
8.60
\begin{equation} \begin{gathered}\label{equation:Hom-in-oriented-fiber-product-reformulated} \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [r]^-{ Q_{+} } \ar [d]^{Q_{-}} & \operatorname{\mathcal{E}}_{+} \ar [d]^{ \pi _{+} } \\ \operatorname{\mathcal{E}}_{-} \ar [r]^-{ \pi _{-} } & \operatorname{Tw}(\operatorname{\mathcal{C}}) } \end{gathered} \end{equation}
is a categorical pullback square (see Notation 8.3.7.3).
Proof.
Using Corollary 4.5.3.24, we can factor the functor $F_{-}$ as a composition $\operatorname{\mathcal{C}}_{-} \xrightarrow { G } \operatorname{\mathcal{C}}'_{-} \xrightarrow { F'_{-} } \operatorname{\mathcal{C}}$, where $G$ is an equivalence of $\infty $-categories and $F'_{-}$ is an isofibration. Set $\operatorname{\mathcal{C}}'_{\pm } = \operatorname{\mathcal{C}}'_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ and define $\operatorname{\mathcal{E}}'_{-} \subseteq \operatorname{Tw}( \operatorname{\mathcal{C}}'_{-} ) \vec{\times }_{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{Tw}(\operatorname{\mathcal{C}})$ as in Notation 8.3.7.3. Then (8.60) can be identified with the outer rectangle in a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [d]^{Q_{-}} \ar [r] & \operatorname{Tw}( \operatorname{\mathcal{C}}'_{\pm } ) \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{+} \ar [d]^{\pi _{+}} \\ \operatorname{\mathcal{E}}_{-} \ar [r] & \operatorname{\mathcal{E}}'_{-} \ar [r] & \operatorname{Tw}(\operatorname{\mathcal{C}}). } \]
It follows from Remark 4.6.4.4 and Corollary 8.1.2.17 that the horizontal map on the upper left is an equivalence of $\infty $-categories. The proof of Lemma 8.3.7.4 shows that the horizontal map on the lower left is also an equivalence of $\infty $-categories, so the left half of the diagram is a categorical pullback square (Proposition 4.5.3.22). Consequently, to show that the outer rectangle is a categorical pullback square, it will suffice to show that the right half of the diagram is a categorical pullback square (Proposition 4.5.3.19). We may therefore replace $\operatorname{\mathcal{C}}_{-}$ by $\operatorname{\mathcal{C}}'_{-}$, and thereby reduce to proving Proposition 8.3.7.6 in the special case where $F_{-}$ is an isofibration of $\infty $-categories. Similarly, we may assume that $F_{+}$ is an isofibration of $\infty $-categories. In this case, the functors $\pi _{-}$ and $\pi _{+}$ are also isofibrations. We are therefore reduced to proving that the functors $Q_{-}$ and $Q_{+}$ induce an equivalence of $\infty $-categories $Q: \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}_{+}$ (see Proposition 4.5.3.27).
The functor $Q$ fits into a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [r]^-{Q} \ar [d] & \operatorname{\mathcal{E}}_{-} \times _{ \operatorname{Tw}(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}_{+} \ar [r] \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{C}}_{-} ) \times \operatorname{Tw}( \operatorname{\mathcal{C}}_{+} ) \ar [d]^{ \operatorname{Tw}(F_{-} ) \times \operatorname{Tw}(F_{+}) } \\ \operatorname{Tw}( \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) ) \ar [r]^-{\overline{Q}} & \operatorname{Fun}_{\pm }( \operatorname{Tw}(\Delta ^1), \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \ar [r] & \operatorname{Tw}(\operatorname{\mathcal{C}}) \times \operatorname{Tw}(\operatorname{\mathcal{C}}), } \]
where $E$ is the equivalence of $\infty $-categories appearing in Example 8.2.4.8. Since the outer rectangle and right half of this diagram are pullback square, the left half of the diagram is also a pullback square. Consequently, to show that $Q$ is an equivalence of $\infty $-categories, it will suffice to show that the vertical maps are isofibrations of $\infty $-categories (Corollary 4.5.3.33). This follows from Corollary 8.1.1.16, by virtue of our assumption that $F_{-}$ and $F_{+}$ are isofibrations.
$\square$