Warning 10.2.0.5. If $\operatorname{\mathcal{C}}$ is an ordinary category, then a simplicial object $X_{\bullet }$ of $\operatorname{\mathcal{C}}$ is completely determined by the collection of objects $\{ X_{n} \} _{n \geq 0}$, together with the face and degeneracy operators
\[ d^{n}_{i}: X_{n} \rightarrow X_{n-1} \quad \quad s^{n}_{i}: X_{n} \rightarrow X_{n+1} \]
(see Proposition 1.1.2.14). In the setting of $\infty $-categories, this is no longer true.