> (Ah, I think the "solution set" and "small cogenerating set" are trivially true if the posets are small... are we assuming that?)
Yes. We're assuming that our preorders and posets are sets: that's what "small" means. So, we haven't been talking about "partially ordered proper classes".
A good example of a partially ordered proper class is the class of all sets, ordered by inclusion. A good example of a totally ordered proper class is the class of ordinals; another nice one is the class of cardinals.
You can get nasty stuff to happen with these. For starters, note that every _subset_ of these posets has a join, but not every _subclass_. For example, the union of any _set_ of sets is a set, but the union of a _class_ of sets may not be a set. Or: every set of ordinals has a least upper bound, but not the class of all ordinals, since the least upper bound of all ordinals would be an ordinal \\(\Omega\\) that's greater than or equal to all others, but we must have \\(\\Omega \le \Omega + 1\\). Similarly, we can't have a largest cardinal, since any cardinal \\(\alpha\\) must have \\(\alpha \le 2^\alpha\\\).
I believe one parlay these problems into an example of a monotone map between partially ordered classes that preserves all _small_ joins but does not have a right adjoint. But I'm not succeeding in inventing one! So I'll record this:
**Puzzle.** Can we find an example of a monotone map between partially ordered classes that preserves all _small_ joins but does not have a right adjoint?
The nLab article you cite very nicely points out the special way in which preorders simplify the conditions on the adjoint functor theorem. The key result, a real shocker, is this:
**Theorem (Freyd).** If a small category has all small limits, or all small colimits, it must be a preorder.
The nLab page [complete small category](https://ncatlab.org/nlab/show/complete+small+category) gives the strikingly simple proof.
The impact of this shocker is that while this theorem is true:
**Theorem.** If \\(C\\) and \\(D\\) are small categories, \\(C\\) has all small limits, and \\(F : C \to D\\) preserves all small limits, \\(F\\) has a left adjoint.
it's of limited usefulness, because these conditions imply that \\(C\\) is a preorder!
So we need a subtler theorem with weaker condition if we want to handle the case when \\(C\\) is a full-fledged category, not a mere preorder.