Fong and Spivak (p.32, Theorem 1.115) give an "adjoint functor theorem for preorders", which I guess is just the adjoint-computing method in this lecture. In addition, Fong and Spivak explicitly say that this method is only applicable when the two preorders **have all meets/joins** (respectively), which John only mentioned in passing:

> In a poset, our desired least upper bound may still not _exist_. But if it does, it's _unique_,...

I think this "applicability condition" is worth emphasizing, because _a_) it is important for beginners to realize and bear in mind that we cannot readily use the formulae (albeit handy) for all preorders, and _b_) the condition is probably generalizable after the notions of limit/colimit are introduced, and it is good for students to realize the conceptual interrelation.