Proposition 7.2.7.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $\operatorname{\mathcal{C}}$ is filtered and $F$ is right cofinal, then $\operatorname{\mathcal{D}}$ is filtered.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We will show that the $\infty $-category $\operatorname{\mathcal{D}}$ satisfies conditions $(a)$ and $(b)$ of Corollary 7.2.6.3. Since $\operatorname{\mathcal{C}}$ is weakly contractible (Proposition 7.2.4.9) and $F$ is a weak homotopy equivalence (Proposition 7.2.1.5), we deduce immediately that $\operatorname{\mathcal{D}}$ is weakly contractible. Suppose we are given left fibrations $U: \widetilde{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{D}}$, $V_{0}: \widetilde{\operatorname{\mathcal{D}}}_{0} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$, and $V_{1}: \widetilde{\operatorname{\mathcal{D}}}_{1} \rightarrow \widetilde{\operatorname{\mathcal{D}}}$, where the $\infty $-categories $\widetilde{\operatorname{\mathcal{D}}}$, $\widetilde{\operatorname{\mathcal{D}}}_{0}$, and $\widetilde{\operatorname{\mathcal{D}}}_{1}$ are weakly contractible. We wish to show that the fiber product $\widetilde{\operatorname{\mathcal{D}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{D}}} } \widetilde{\operatorname{\mathcal{D}}}_{1}$ is also weakly contractible. Set $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \widetilde{\operatorname{\mathcal{D}}}$, and define $\widetilde{\operatorname{\mathcal{C}}}_{0}$ and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ similarly. Applying Proposition 7.2.3.12, we deduce that the projection maps
\[ \widetilde{\operatorname{\mathcal{C}}}_{0} \rightarrow \widetilde{\operatorname{\mathcal{D}}}_{0} \quad \quad \widetilde{\operatorname{\mathcal{C}}} \rightarrow \widetilde{\operatorname{\mathcal{D}}} \quad \quad \widetilde{\operatorname{\mathcal{C}}}_{1} \rightarrow \widetilde{\operatorname{\mathcal{D}}}_{1} \]
are right cofinal; in particular, they are weak homotopy equivalences (Proposition 7.2.4.9). It follows that the $\infty $-categories $\widetilde{\operatorname{\mathcal{C}}}$, $\widetilde{\operatorname{\mathcal{C}}}_{0}$, and $\widetilde{\operatorname{\mathcal{C}}}_{1}$ are weakly contractible. Since $\operatorname{\mathcal{C}}$ is filtered, Corollary 7.2.6.3 guarantees that the fiber product $\widetilde{\operatorname{\mathcal{C}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1}$ is weakly contractible. The projection map
\[ \widetilde{\operatorname{\mathcal{C}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{C}}} } \widetilde{\operatorname{\mathcal{C}}}_{1} \rightarrow \widetilde{\operatorname{\mathcal{D}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{D}}} } \widetilde{\operatorname{\mathcal{D}}}_{1} \]
is also right cofinal (Proposition 7.2.3.12) and therefore a weak homotopy equivalence (Proposition 7.2.4.9). It follows that $\widetilde{\operatorname{\mathcal{D}}}_{0} \times _{ \widetilde{\operatorname{\mathcal{D}}} } \widetilde{\operatorname{\mathcal{D}}}_{1}$ is also weakly contractible, as desired. $\square$