Lemma 9.1.5.4. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category. Suppose we are given left fibrations of $\infty $-categories $U: \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$, $V_0: \widetilde{\operatorname{\mathcal{C}}}_0 \rightarrow \widetilde{\operatorname{\mathcal{C}}}$, and $V_{1}: \widetilde{\operatorname{\mathcal{C}}}_{1}$. If $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_0$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are weakly contractible, then the fiber product $\widetilde{\operatorname{\mathcal{C}}}_0 \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1}$ is also weakly contractible.
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Proof. It follows from Theorem 9.1.3.2 that the $\infty $-categories $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_{0}$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are filtered. Applying Lemma 9.1.5.2, we conclude that the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}_0 \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1}$ is also filtered, and therefore weakly contractible (Proposition 9.1.1.13). $\square$