Let \(P \overset{f}{\underset{g}{\rightleftarrows}} Q\) be monotone maps with \(f \dashv g\).
Right adjoints preserve meets
Left adjoints preserve joins
Given \(A \subseteq P\) and its image \(f(A) \subseteq Q\)
Then, if \(A\) has a join \(\vee A \in P\), then \(\vee f(a) \in Q\) exists
And \(f(\vee A) \cong \vee f(A)\)
Left adjoints preserve joins
let \(j = \vee A \subseteq P\)
Given f is monotone, \(\forall a \in A: f(a) \leq f(j)\), i.e. we have \(f(a)\) as an upper bound for \(f(A)\)
To show it is a least upper bound, take some arbitrary other upper bound b for \(f(A)\) and show that \(f(j) \leq b\)
Because \(j\) is the least upper bound of \(A\), we have \(j \leq g(b)\)
Using the Galois connection, we have \(f(j) \leq b\) showing that \(f(j)\) is the least upper bound of \(f(A) \subseteq Q\).
Right adjoints preserving meets is dual to this.