Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation $\sim$ on a set $A$ is antisymmetric if any two elements that are related in both orders are equal:
In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is antisymmetric if its intersection with its reverse is contained in the identity relation on $A$:
If an antisymmetric relation is also reflexive (as most are in practice), then this containment becomes an equality.