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Corollary 9.1.4.15. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be a $\kappa $-filtered $\infty $-category. For every $\kappa $-small diagram $q: K \rightarrow \operatorname{\mathcal{C}}$, the forgetful functor $U: \operatorname{\mathcal{C}}_{q/} \rightarrow \operatorname{\mathcal{C}}$ is right cofinal.

Proof. It follows from Proposition 9.1.1.16 that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{q/}$ is $\kappa $-filtered, so that $U$ is right cofinal by virtue of Corollary 9.1.4.14. $\square$