Rel, bicategory of relations, allegory
left and right euclidean;
extensional, well-founded relations.
A (binary) relation $\sim$ on a set $A$ is asymmetric if no two elements are related in both orders:
In the language of the $2$-poset-with-duals Rel of sets and relations, a relation $R: A \to A$ is asymmetric if it is disjoint from its dual:
Of course, this containment is in fact an equality.
An asymmetric relation is necessarily irreflexive.