Remark 1.3.1.7 (Face Operators on $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$). Let $\operatorname{\mathcal{C}}$ be a category and suppose we are given an $n$-simplex $\sigma $ of the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ for some $n > 0$, which we identify with a diagram
Then:
The $0$th face $d^{n}_0(\sigma ) \in \operatorname{N}_{n-1}(\operatorname{\mathcal{C}})$ can be identified with the diagram
\[ C_1 \xrightarrow {f_2} C_2 \xrightarrow {f_3} C_3 \rightarrow \cdots \xrightarrow {f_ n} C_ n \]obtained from $\sigma $ by “deleting” the object $C_0$ (and the morphism $f_1$ with source $C_0$).
The $n$th face $d^{n}_ n(\sigma ) \in \operatorname{N}_{n-1}(\operatorname{\mathcal{C}})$ can be identified with the diagram
\[ C_0 \xrightarrow {f_1} C_1\rightarrow \cdots \rightarrow C_{n-2} \xrightarrow { f_{n-1} } C_{n-1} \]obtained from $\sigma $ by “deleting” the object $C_ n$ (and the morphism $f_ n$ with target $C_ n$).
For $0 < i < n$, the $i$th face $d^{n}_ i(\sigma ) \in \operatorname{N}_{n-1}(\operatorname{\mathcal{C}})$ can be identified with the diagram
\[ C_0 \xrightarrow {f_1} C_1 \rightarrow \cdots \rightarrow C_{i-1} \xrightarrow { f_{i+1} \circ f_ i} C_{i+1} \rightarrow \cdots \xrightarrow {f_ n} C_ n \]obtained by “deleting” the object $C_{i}$ (and composing the morphisms $f_{i}$ and $f_{i+1}$).