addendum: The situation for meets is a bit more complicated. \\(\downarrow\\) preserves meets but not joins, so it doesn't have a right adjoint. Instead we have to reverse the ordering on \\(PA\\), then we get \\(\operatorname{\uparrow}a \supseteq S\\) iff \\(a \le \bigwedge S\\) , and so \\(\bigwedge\\) is right adjoint to \\(\uparrow\\).