Let ${f : E \rightarrow S}$ be a geometric morphism (between toposes).


For $s$ in $S$ and $x$ in $E$ let ${\pi : f^\* s \times x \rightarrow x}$ be the obvious projection in $E$.


Let ${u \rightarrow f^\* s \times x}$ be a complemented subobject of ${f^\* s \times x}$.


Is the image of $u$ along $\pi$ complemented as a subobject of $x$?


(See also [Images of complemented subobjects in hyperconnected toposes over Boolean bases](https://mathoverflow.net/questions/396041/images-of-complemented-subobjects-in-hyperconnected-toposes-over-boolean-bases))



Images of complemented subobjects in toposes
No, not even if $E=S$, $f$ is the identity morphism, and $x=1$. In that special case, your question asks whether $\forall z\in s\,\big((z\in u)\lor \neg(z\in u)\big)$ (in the internal language of $S$) implies $(\exists z\in s\,z\in u)\lor\neg(\exists z\in s\,z\in u)$. When $s$ is $\mathbb N$, this is the limited principle of omniscience, which is not intuitionistically valid.


