Remark 10.2.5.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Stated more informally, an augmented simplicial object $C_{\bullet }$ of $\operatorname{\mathcal{C}}$ is a Čechnerve if, for every integer $n \geq 0$, it exhibits $C_{n}$ as an iterated fiber product
\[ C_{0} \times _{ C_{-1} } C_{0} \times _{ C_{-1} } \cdots \times _{ C_{-1} } C_{0} \]
(where the factor $C_0$ appears $n+1$ times).