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 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{colim}}\webleft (\webleft (f\mathbin {\overset {\to }{\times }}\underline{\webleft (-_{1}\webright )}\webright )\overset {\text{pr}}{\twoheadrightarrow }A\overset {\chi _{U}}{\to }\{ \mathsf{t},\mathsf{f}\} \webright )\\ & = \operatorname*{\text{colim}}_{\substack {x\in X\\ f\webleft (x\webright )=-_{1}
}}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & = \bigvee _{\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 )& =\bigvee _{\substack {x\in X\\ f\webleft (x\webright )=y
}}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & =\begin{cases} \mathsf{true}& \text{if there exists some $x\in X$ such}\\ & \text{that $f\webleft (x\webright )=y$ and $x\in U,$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & =\begin{cases} \mathsf{true}& \text{if there exists some $x\in U$}\\ & \text{such that $f\webleft (x\webright )=y$,}\\ \mathsf{false}& \text{otherwise} \end{cases}\end{align*}
for each $y\in Y$.
