= Solution
The defining square for the <subobject classifier> is a <pullback in a category>:
$$
\begin{array}{ccc}A&\longrightarrow&1\\\downarrow&&\downarrow\top_{\mathcal E}\\X&\xrightarrow{a}&\Omega_{\mathcal E}.\end{array}
$$
The <inverse image functor of a geometric morphism> preserves this square and the terminal object. By definition, the inverse-image truth arrow $f^*\top_{\mathcal E}$ is itself the pullback of $\top_{\mathcal F}$ along $\bar\lambda$. Composing these two pullback squares shows that $f^*A\hookrightarrow f^*X$ is the truth pullback along $\bar\lambda\circ f^*a$. Uniqueness of the characteristic map gives
$$
\boxed{\chi_{f^*A}=\bar\lambda\circ f^*a}.
$$
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