Corollary 9.1.4.18. Let $\kappa $ be a regular cardinal, let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\kappa $-filtered $\infty $-categories, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a right cofinal functor. Then, for every $\kappa $-small diagram $q: K \rightarrow \operatorname{\mathcal{C}}$, the functor of coslice $\infty $-categories $F_{q/}: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{D}}_{ (F \circ q)/}$ is also right cofinal.
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Proof. Let $U: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{C}}$ and $V: \operatorname{\mathcal{D}}_{ (F \circ q)/} \rightarrow \operatorname{\mathcal{D}}$ be the forgetful functors. Since $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, the left fibration $U$ is right cofinal (Corollary 9.1.4.15). Applying Proposition 9.1.4.17, we conclude that $F \circ U = V \circ F_{q/}$ is also right cofinal. Since $V$ is also left fibration, Proposition 9.1.4.17 guarantees that $F_{q/}$ is right cofinal. $\square$