Example 4.6.4.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(Q, \leq )$ be a partially ordered set and let $q \in Q$ be a maximal element. Set
\[ Q_{\leq q} = \{ p \in Q: p \leq q \} \quad \quad Q_{< q} = \{ p \in Q: p < q \} \quad \quad Q_0 = \{ q \in Q: p \neq q \} . \]
Then:
The nerve $\operatorname{N}_{\bullet }( Q_{\leq q} )$ can be identified with the right cone $\operatorname{N}_{\bullet }( Q_{< q} )^{\triangleright }$. Applying Example 4.6.4.9, we obtain an equivalence of $\infty $-categories
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }(Q_{\leq q} ), \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( \operatorname{N}_{\bullet }(Q_{< q} ), \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{N}_{\bullet }(Q_{< q} ), \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}}. \]The diagram of inclusion maps
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }(Q_{is a pushout square, and therefore induces an isomorphism of $\infty $-categories
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }(Q), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }(Q_0), \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( Q_{Applying Corollary 4.5.2.29, we obtain an equivalence of $\infty $-categories
\[ \operatorname{Fun}( \operatorname{N}_{\bullet }(Q), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }(Q_0), \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{N}_{\bullet }(Q_{