Identifying $\mathcal{P}\webleft (Y\webright )$ with $\mathsf{Sets}\webleft (Y,\{ \mathsf{t},\mathsf{f}\} \webright )$ via Item 2 of Proposition 2.5.1.1.4, we see that the inverse image function associated to $f$ is equivalently the function
\[ f^{*}\colon \mathcal{P}\webleft (Y\webright )\to \mathcal{P}\webleft (X\webright ) \]
defined by
\[ f^{*}\webleft (\chi _{V}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{V}\circ f \]
for each $\chi _{V}\in \mathcal{P}\webleft (Y\webright )$, where $\chi _{V}\circ f$ is the composition
\[ X\overset {f}{\to }Y\overset {\chi _{V}}{\to }\{ \mathsf{true},\mathsf{false}\} \]
in $\mathsf{Sets}$.
