Christian wrote:

> (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.