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Proposition 10.2.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čech nerve $\operatorname{\check{C}}(X/Y)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} ) \rightarrow \operatorname{\mathcal{C}}$ (see Definition 10.1.5.4). Then $f$ is a quotient morphism if and only if $\operatorname{\check{C}}_{\bullet }(X/Y)$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Proof of Proposition 10.2.2.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$ which admits a Čech nerve $\operatorname{\check{C}}(X/Y)_{\bullet }: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}}_{+} ) \rightarrow \operatorname{\mathcal{C}}$. We wish to show that $f$ is a quotient morphism if and only if $\operatorname{\check{C}}(X/Y)_{\bullet }$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{C}}^{0}_{/Y} \subseteq \operatorname{\mathcal{C}}_{/Y}$ denote the sieve generated by $X$ and let $Q$ denote the composite map

\[ (\operatorname{\mathcal{C}}^{0}_{/Y})^{\triangleright } \hookrightarrow (\operatorname{\mathcal{C}}_{/Y})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}. \]

Let us identify $\operatorname{\check{C}}(X/Y)_{\bullet }$ with a functor $F: \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \rightarrow \operatorname{\mathcal{C}}_{/Y}^{\operatorname{op}}$. Unwinding the definitions, we wish to show that $Q$ is a colimit diagram if and only if the composite functor

\[ \operatorname{N}_{\bullet }( \operatorname{{\bf \Delta }}^{\operatorname{op}} )^{\triangleright } \xrightarrow { F^{\triangleright } } (\operatorname{\mathcal{C}}^{0}_{/Y} )^{\triangleright } \xrightarrow {Q} \operatorname{\mathcal{C}} \]

is a colimit diagram. This is a special case of Corollary 7.2.2.3, since the functor $F$ is right cofinal (Variant 10.2.2.6). $\square$