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Corollary 7.6.4.9. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, let let $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ denote the homotopy fiber product of Construction 4.5.2.1, and let

\[ G_0: \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_0 \quad \quad G_1: \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_1 \]

denote the projection maps, so that we have a canonical isomorphism $\alpha : F_0 \circ G_0 \rightarrow F_1 \circ G_1$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1, \operatorname{\mathcal{C}})$. Then the diagram

\[ \xymatrix@C =100pt@R=100pt{ \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1 \ar [r]^-{G_0} \ar [d]_-{G_1} \ar [dr]^-{F_1 \circ G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^-{F_0} \ar@ {=>}[]+<-20pt,-20pt>;+<-45pt,-45pt>_-{\alpha } \\ \operatorname{\mathcal{C}}_1 \ar [r]_-{ F_1 } \ar@ {=>}[]+<20pt,20pt>;+<45pt,45pt>^-{\operatorname{id}} & \operatorname{\mathcal{C}}} \]

corresponds to a pullback square in the $\infty $-category $\operatorname{\mathcal{QC}}$. In particular, $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a fiber product of $\operatorname{\mathcal{C}}_0$ with $\operatorname{\mathcal{C}}_1$ over $\operatorname{\mathcal{C}}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$.

Proof. By virtue of Proposition 7.6.4.8, it will suffice to show that the inclusion

\[ \delta : \operatorname{\mathcal{C}}_{1} \simeq \operatorname{\mathcal{C}}_1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}_{1} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}} \]

induces an equivalence of homotopy fiber products

\[ \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1} \hookrightarrow \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} ( \operatorname{\mathcal{C}}_1 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}). \]

This is a special case of Corollary 4.5.2.18, since $\delta $ is an equivalence of $\infty $-categories (Proposition 5.3.7.4). $\square$