Identifying subsets of $A$ with functions from $A$ to $\{ \mathsf{true},\mathsf{false}\} $ via Item 1 and Item 2 of Proposition 2.4.3.1.6, we see that the direct image with compact support function associated to $f$ is equivalently the function
\[ f_{!}\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]
defined by
\begin{align*} f_{!}\webleft (\chi _{U}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ran}_{f}\webleft (\chi _{U}\webright )\\ & = \operatorname*{\text{lim}}\webleft (\webleft (\underline{\webleft (-_{1}\webright )}\mathbin {\overset {\to }{\times }}f\webright )\overset {\text{pr}}{\twoheadrightarrow }A\overset {\chi _{U}}{\to }\{ \mathsf{true},\mathsf{false}\} \webright )\\ & = \operatorname*{\text{lim}}_{\substack {a\in A\\ f\webleft (a\webright )=-_{1}
}}\webleft (\chi _{U}\webleft (a\webright )\webright )\\ & = \bigwedge _{\substack {a\in A\\ f\webleft (a\webright )=-_{1}
}}\webleft (\chi _{U}\webleft (a\webright )\webright ).\end{align*}
where we have used 
for the second equality. In other words, we have
\begin{align*} \webleft [f_{!}\webleft (\chi _{U}\webright )\webright ]\webleft (b\webright )& =\bigwedge _{\substack {a\in A\\ f\webleft (a\webright )=b
}}\webleft (\chi _{U}\webleft (a\webright )\webright )\\ & =\begin{cases} \mathsf{true}& \text{if, for each $a\in A$ such that}\\ & \text{$f\webleft (a\webright )=b$, we have $a\in U,$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & =\begin{cases} \mathsf{true}& \text{if $f^{-1}\webleft (b\webright )\subset U$}\\ \mathsf{false}& \text{otherwise} \end{cases}\end{align*}
for each $b\in B$.
