Remark 1.1.5.3. Let $C$ be an object of the category $\operatorname{\mathcal{C}}$. The constant simplicial object $\underline{C}_{}$ can be described concretely as follows:
For each $n \geq 0$, we have $\underline{C}_{n} = C$.
The face and degeneracy operators
\[ d^{n}_{i}: \underline{C}_{n} \rightarrow \underline{C}_{n-1} \quad \quad s^{n}_{i}: \underline{C}_{n} \rightarrow \underline{C}_{n+1} \]are the identity maps from $C$ to itself.