Given \(f \dashv g\), prove that the composition of left and right adjoint monotone maps is a closure operator
Show \(p \leq (f;g)(p)\)
Show \((f;g;f;g)(p) \cong (f;g)(p)\)
This is one of the conditions of adjoint functors: \(p \leq g(f(p))\)
The \(\leq\) direction is an extension of the above: \(p \leq g(f(p)) \leq g(f(g(f(p))))\)
Galois property: \(q \geq f(g(q))\), substitute \(f(p)\) for \(q\) to get \(f(p) \geq f(g(f(p)))\).
Because \(g\) is a monotone map, we can apply it to both sides to get \(g(f(p)) \geq g(f(g(f(p))))\)