Proposition 9.1.4.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cocartesian fibration of $\infty $-categories. If $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are filtered, then $F$ is right cofinal.
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Proof. For each object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{D}}_{D/}$ is filtered (Proposition 9.1.1.17) and the projection map $\operatorname{\mathcal{D}}_{D/} \rightarrow \operatorname{\mathcal{D}}$ is a cocartesian fibration (Proposition 4.3.6.1). Applying Corollary 9.1.3.14, we conclude that the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{D/}$ is a filtered $\infty $-category, and is therefore weakly contractible (Proposition 9.1.1.18). Allowing the object $D$ to vary, we conclude that the functor $F$ is right cofinal (Theorem 7.2.3.1). $\square$