The symmetry of $\mathsf{Rel}$ is the natural isomorphism
\[ \sigma ^{\mathsf{Rel}}_{A,B} \colon A\times B \to B\times A \]
at $(A,B)$ is defined by declaring
\[ (a,b) \sim _{\sigma ^{\mathsf{Rel}}_{A,B}} (b',a') \]
iff $a=a'$ and $b=b'$.
