Adjoints of inverse image on power sets (${\exists }_p⊣p^{\ast }⊣{\forall }_p$)

For a function $p:A\to B$, order each power set by inclusion. The inverse image map $p^{\ast }(V)=p^{-1}(V)$ has a left adjoint ${\exists }_p(U)=p(U)$ and a right adjoint ${\forall }_p(U)=\{b\in B:p^{-1}(\{b\})\subseteq U\}$. The two adjunctions follow from $p(U)\subseteq V⟺U\subseteq p^{-1}(V)$ and $p^{-1}(V)\subseteq U⟺V\subseteq {\forall }_p(U)$. Empty fibers satisfy the second condition vacuously, so points outside the image of $p$ belong to ${\forall }_p(U)$ for every $U$.