Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 23 / 5 / b / Solution

For $X\subseteq A$, defineBoth are monotone maps $PA\to PB$. For $Y\subseteq B$,so direct image is a left adjoint to inverse image. Also,so the fiberwise universal condition gives the right adjoint. ThereforeAn empty fiber satisfies the universal condition vacuously, so every point outside $p[A]$ belongs to ${\forall }_p(X)$, regardless of $X$. This is essential when $p$ is not surjective.