Remark 1.4.2.3. Let $S_{\bullet }$ be a simplicial set. Then the opposite simplicial set $S_{\bullet }^{\operatorname{op}}$ can be described more concretely as follows:
For each $n \geq 0$, we have $S_{n}^{\operatorname{op}} = S_{n}$.
The face and degeneracy operators of $S_{\bullet }^{\operatorname{op}}$ are given by
\[ (d^{n}_ i: S^{\operatorname{op}}_ n \rightarrow S^{\operatorname{op}}_{n-1}) = (d^{n}_{n-i}: S_ n \rightarrow S_{n-1}) \]\[ (s^{n}_ i: S^{\operatorname{op}}_ n \rightarrow S^{\operatorname{op}}_{n+1}) = (s^{n}_{n-i}: S_ n \rightarrow S_{n+1}). \]