Exercise 10.2.1.6. Let $X_{\bullet }$ be a simplicial object of an ordinary category $\operatorname{\mathcal{C}}$. Show that an object $X \in \operatorname{\mathcal{C}}$ is a geometric realization of $X_{\bullet }$ (in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$) if and only if it is a coequalizer of the face operators $d^{1}_0, d^{1}_1: X_1 \rightrightarrows X_0$. For a slightly more general statement, see Corollary 10.2.2.12.
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