Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation $\sim$ on a set $A$ is symmetric if any two elements that are related in one order are also related in the other order:
In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is symmetric if it is contained in its reverse:
In that case, this containment is in fact an equality.
symmetric relation