Exercise 1.1.2.7 (Relations Between Face and Degeneracy Operators). Let $C_{\bullet }$ be a simplicial object of a category $\operatorname{\mathcal{C}}$. Show that the face and degeneracy operators of $\operatorname{\mathcal{C}}$ satisfy the following relations:
- $(\ast ')$
For $0 \leq i \leq n$ and $0 \leq j \leq n+1$, we have an equality
\[ d^{n+1}_{j} \circ s^{n}_ i = \begin{cases} s^{n-1}_{i-1} \circ d^{n}_ j & \text{ if } j < i \\ \operatorname{id}_{ C_ n } & \text{ if } j = i \text{ or } j = i + 1 \\ s^{n-1}_{i} \circ d^{n}_{j-1} & \text{ if } j > i+1 \end{cases} \](as morphisms from $C_{n}$ to $C_{n}$).