<a’,b’> ⊆ <a,b> if a ⊆’ a and b ⊆’ b.
Order-preserving maps can be defined each way between these two partial orders. From ζ(U) to ζ(U) x ζ(U), there is the diagonal map Δ(x) = <x,x>, and from ζ(U) x ζ(U) to ζ(U), there is the meet map ∩(<a,b>) = a ∩ b. Consider now the following “adjointness relation” between the two partial orders:
Δ(c) ⊆ <a,b> iff c ⊆ ∩ (<a,b>) Adjointness Equivalence
for sets a, b, and c in ζ(U). It has a certain symmetry that can be exploited. If we fix <a,b>, then we have the previous universality condition for the meet of a and b: for any c in ζ(U), c ⊆ a ∩ b iff Δ(c) ⊆ <a,b> Universality Condition for Meet of Sets a and b.
The defining property on elements c of ζ(U) is that Δ(c) ⊆ <a,b>. But using the symmetry, we could fix c and have another universality condition using the reverse inclusion in ζ(U) x ζ(U) as the participation relation: for any <a,b> in ζ(U) x ζ(U), <a,b> ⊇ Δ(c) iff c ⊆ a ∩ b. Universality Condition for Δ(c). Here the defining property on elements <a,b> of ζ(U) x ζ(U) is that “the meet of a and b is a superset of the given set c.” The self-predicative universal for that property is the image of c under the diagonal map Δ(c) = <c,c>, just as the self-predicative universal for the other property defined given <a,b> was the image of <a,b> under the meet map ∩(<a,b>) = a ∩ b.
Thus in this adjoint situation between the two categories ζ(U) and ζ(U) x ζ(U), we have a pair of maps (“adjoint functors”) going each way between the categories such that each element in a category defines a certain property in the other category and the map carries the element to the self-predicative universal for that property.
Δ: ζ(U) → ζ(U) x ζ(U) and ∩: ζ(U) x ζ(U) → ζ(U) Example of Adjoint Functors Between Partial Orders
The notion of a pair of adjoint functors is ubiquitous; it is one of the main tools that highlights self-predicative universals throughout modern mathematics.