Example 9.1.1.8. Let $\operatorname{Idem}$ be the category introduced in Construction 8.5.2.7, comprised of a single object $X$ and a single non-identity morphism $e: X \rightarrow X$ satisfying $e \circ e = e$. Then any diagram $f: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{Idem})$ admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Idem})$, characterized by the requirement that for each vertex $k \in K$, the composition
\[ \Delta ^1 \simeq \{ k\} ^{\triangleright } \hookrightarrow K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{N}_{\bullet }( \operatorname{Idem}) \]
is the morphism $e$. It follows that the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{Idem})$ is $\kappa $-filtered for every infinite cardinal $\kappa $ (see Exercise 8.5.2.9).