Theorem 8.3.7.1. Let $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(\bullet , \bullet ): \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{\mathcal{C}}$, and define $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{-} }(\bullet , \bullet )$, $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_+}(\bullet , \bullet )$, and $\operatorname{Hom}_{ \operatorname{\mathcal{C}}_{\pm } }(\bullet , \bullet )$ similarly. Then there is a pullback diagram
8.58
\begin{equation} \begin{gathered}\label{equation:Hom-in-oriented-fiber-product} \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{\pm }}( \bullet , \bullet ) \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{+} }( \operatorname{ev}_{+}( \bullet ), \operatorname{ev}_{+}( \bullet ) ) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}_{-}}( \operatorname{ev}_{-}(\bullet ), \operatorname{ev}_{-}(\bullet ) ) \ar [r] & \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( (F_{-} \circ \operatorname{ev}_-)(\bullet ), (F_{+} \circ \operatorname{ev}_+)(\bullet ) ) } \end{gathered} \end{equation}
in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{\pm }, \operatorname{\mathcal{S}})$.
Proof of Theorem 8.3.7.1.
Let $U_{\pm }: \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \rightarrow \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{\pm }$ and $U: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}$ be the projection maps of Notation 8.1.1.6. We have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Tw}( \operatorname{\mathcal{C}}_{\pm } ) \ar [rr]^{ Q_{+} } \ar [dr]^{ U_{\pm } } \ar [dd]^{Q_{-}} & & \operatorname{\mathcal{E}}_{+} \ar [dr]^{ U_{+} } \ar [dd] & \\ & \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \times \operatorname{\mathcal{C}}_{\pm } \ar [rr] \ar [dd] & & ( \operatorname{\mathcal{C}}_{+}^{\operatorname{op}} \vec{\times }_{ \operatorname{\mathcal{C}}^{\operatorname{op}} } \operatorname{\mathcal{C}}^{\operatorname{op}} ) \times \operatorname{\mathcal{C}}_{+} \ar [dd] \\ \operatorname{\mathcal{E}}_{-} \ar [rr] \ar [dr]^{ U_{-} } & & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [dr]^{U} & \\ & \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \times (\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}) \ar [rr] & & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}, } \]
where the diagonal maps are left fibrations whose covariant transport representations are given by (the restrictions of) $\operatorname{Hom}$-functors for the $\infty $-categories $\operatorname{\mathcal{C}}$, $\operatorname{\mathcal{C}}_{-}$, $\operatorname{\mathcal{C}}_{+}$, and $\operatorname{\mathcal{C}}_{\pm }$ (Lemma 8.3.7.4). By virtue of Lemma 8.3.7.7, it will suffice to show that the front and back faces of this diagram are categorical pullback squares. For the back face, this follows from Proposition 8.3.7.6. The front face is a cartesian product of a pair of pullback diagrams
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{\pm }^{\operatorname{op}} \ar [r] \ar [d] & \operatorname{\mathcal{C}}^{\operatorname{op}}_{+} \vec{\times }_{\operatorname{\mathcal{C}}^{\operatorname{op}}} \operatorname{\mathcal{C}}^{\operatorname{op}} \ar [d] & \operatorname{\mathcal{C}}_{\pm } \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{+} \ar [d] \\ \operatorname{\mathcal{C}}_{-}^{\operatorname{op}} \ar [r] & \operatorname{\mathcal{C}}^{\operatorname{op}} & \operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}. } \]
We are therefore reduced to showing that these pullback squares are categorical pullback squares (Remark 4.5.3.10). This follows from Corollary 4.5.3.28, since vertical maps of the left square and horizontal maps of the right square are isofibrations (Proposition 4.6.4.2).
$\square$