Identifying subsets of $B$ with functions from $B$ to $\{ \mathsf{true},\mathsf{false}\} $ via Item 1 and Item 2 of Proposition 2.4.3.1.6, we see that the inverse image function associated to $f$ is equivalently the function
\[ f^{*}\colon \mathcal{P}\webleft (B\webright )\to \mathcal{P}\webleft (A\webright ) \]
defined by
\[ f^{*}\webleft (\chi _{V}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\chi _{V}\circ f \]
for each $\chi _{V}\in \mathcal{P}\webleft (B\webright )$, where $\chi _{V}\circ f$ is the composition
\[ A\overset {f}{\to }B\overset {\chi _{V}}{\to }\{ \mathsf{true},\mathsf{false}\} \]
in $\mathsf{Sets}$.
