Identifying $\mathcal{P}\webleft (X\webright )$ with $\mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )$ via Item 2 of Proposition 2.5.1.1.4, we see that the direct image with compact support function associated to $f$ is equivalently the function
\[ f_{!}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright ) \]
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 }X\overset {\chi _{U}}{\to }\{ \mathsf{true},\mathsf{false}\} \webright )\\ & = \operatorname*{\text{lim}}_{\substack {x\in X\\ f\webleft (x\webright )=-_{1}
}}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & = \bigwedge _{\substack {x\in X\\ f\webleft (x\webright )=-_{1}
}}\webleft (\chi _{U}\webleft (x\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 (y\webright ) & = \bigwedge _{\substack {x\in X\\ f\webleft (x\webright )=y
}}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & = \begin{cases} \mathsf{true}& \text{if, for each $x\in X$ such that}\\ & \text{$f\webleft (x\webright )=y$, we have $x\in U,$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & = \begin{cases} \mathsf{true}& \text{if $f^{-1}\webleft (y\webright )\subset U$}\\ \mathsf{false}& \text{otherwise} \end{cases}\end{align*}
for each $y\in Y$.
