Example 1.4.2.5. Let $X$ be a topological space. Then there is a canonical isomorphism of simplicial sets $\operatorname{Sing}_{\bullet }(X) \simeq \operatorname{Sing}_{\bullet }(X)^{\operatorname{op}}$, which carries each singular $n$-simplex $\sigma : | \Delta ^{n} | \rightarrow X$ to the composite map
\[ | \Delta ^{n} | \xrightarrow {r} | \Delta ^ n | \xrightarrow {\sigma } X \]
where $r$ is denotes the homeomorphism of $| \Delta ^ n |$ with itself given by $r( t_0, t_1, \ldots , t_{n-1}, t_ n) = (t_ n, t_{n-1}, \ldots , t_1, t_0)$.