Proposition 9.1.1.12. Let $\kappa $ be an infinite cardinal, let $\operatorname{\mathcal{C}}$ be a $\kappa $-filtered $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, where $K$ is a $\kappa $-small simplicial set. Then the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is also $\kappa $-filtered.
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Proof. By virtue of Remark 9.1.1.10, it will suffice to show that for every $\kappa $-small simplicial set $L$ and every morphism $g: L \rightarrow \operatorname{\mathcal{C}}_{f/}$, the $\infty $-category $(\operatorname{\mathcal{C}}_{f/} )_{g/}$ is nonempty. Unwinding the definitions, we can identify $g$ with a morphism of simplicial sets $\overline{f}: K \star L \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{f}|_{K} = f$. This identification supplies an isomorphism $(\operatorname{\mathcal{C}}_{f/} )_{g/} \simeq \operatorname{\mathcal{C}}_{ \overline{f} / }$. We are therefore reduced to showing that the coslice $\infty $-category $\operatorname{\mathcal{C}}_{ \overline{f} / }$ is nonempty. This follows from Remark 9.1.1.10, since the simplicial set $K \star L$ is $\kappa $-small (Corollary 4.7.4.13). $\square$