This is a follow-up question from this question of mine.
In the same paper as the one mentioned in my previous post, it's stated that
In the context of complete lattices, a monotone map has a right adjoint if and only if it preserves all joins and a monotone map has a left adjoint if and only if it preserves all meets.
I know that $\sup$ & $\inf$ can each be written in terms of join and meet, so I can sort of see why this is true, but how can I see it categorically?
Please help :)
It is an immediate consequence of Freyd's general adjoint functor theorem. The solution set condition is empty for small categories, in particular for small preorders. This is explained very well in the nlab.