Page "Concrete category" Paragraph 16

# The category Rel whose objects are sets and whose morphisms are relations can be made concrete by taking U to map each set X to its power set and each relation to the function defined by.

Noting that power sets are complete lattices under inclusion, those functions between them arising from some relation R in this way are exactly the supremum-preserving maps.

Hence Rel is equivalent to a full subcategory of the category Sup of complete lattices and their sup-preserving maps.

Conversely, starting from this equivalence we can recover U as the composite Rel → Sup → Set of the forgetful functor for Sup with this embedding of Rel in Sup.