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Exercise Let $C_{\bullet }$ be a semisimplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face maps of Notation satisfy the following condition:

$(\ast )$

For $n \geq 2$ and $0 \leq i < j \leq n$, we have $d_{i} \circ d_{j} = d_{j-1} \circ d_{i}$ (as a map from $C_{n}$ to $C_{n-2}$).

Conversely, show that any collection of objects $\{ C_ n \} _{n \geq 0}$ and morphisms $\{ d_ i: C_ n \rightarrow C_{n-1} \} _{0 \leq i \leq n}$, satisfying $(\ast )$ determines a unique semisimplicial object of $\operatorname{\mathcal{C}}$.