Kerodon

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Proposition 10.2.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pullbacks. Then every morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ admits a Čechnerve $\operatorname{\check{C}}_{\bullet }(X/Y)$.

Proof. Apply Corollary 10.2.4.6 to the $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$, which admits finite products by virtue of our assumption that $\operatorname{\mathcal{C}}$ admits pullbacks (Corollary 7.6.2.17). $\square$