= Adjoints of inverse image on power sets
{title2=$\exists_p\dashv p^*\dashv\forall_p$}
For a <function> $p:A\to B$, order each <power set> by inclusion. The <inverse image> map $p^*(V)=p^{-1}(V)$ 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 $p(U)\subseteq V\iff U\subseteq p^{-1}(V)$ and $p^{-1}(V)\subseteq U\iff V\subseteq\forall_p(U)$. Empty <fibers> satisfy the second condition vacuously, so points outside the image of $p$ belong to $\forall_p(U)$ for every $U$.
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