Poset-enriched adjunction

In a category whose hom-sets are posets, $f:A\to B$ is left adjoint to $g:B\to A$ when $fg\le 1_B$ and $1_A\le gf$. Left adjoints are closed under identities and composition.

In the inclusion-ordered category of sets and relations, a relation has a right adjoint exactly when it is the graph of a total single-valued function; its right adjoint is the converse relation.