Images of complemented subobjects in hyperconnected toposes over Boolean bases Let $S$ be a Boolean topos.
Let ${f : E \rightarrow S}$ be a hyperconnected geometric morphism.
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 $\pi$ a complemented subobject of $x$?
(Note: this is a continuation of Images of complemented subobjects in toposes)
 A: No.
Take $S$ to be Sets, then for any set $s$, $f^* s \times x$ is the coproduct of $s$-copies of $X$, and a complemented subobject of $f^* s \times x$  is the same as an $s$ indexed collection of complemented subobject of $x$. The the image by the projection $f^*s \times x \to x$ is their union.
So, for $S= Set$, your question translate into: Are complemented subobject closed under union in hyperconnected topos ? This is not the case.
For example, the Jonsson-Tarski topos is an hyperconnected topos, which has a slice equivalent to Sh(K) for K the Cantor space. In particular it has an object $X$ such that the subobject of $X$ identifies with the open subset of the Cantor space and every open subset of the Cantor is a union of clopen (i.e. complemented open)
Edit: It was asked in the comment to give a description of the left adjoint $f^*$ from Sets to JT of the global section functor $f_*$ topos. It happens that there is a fairly explicit description of it. The trick is that the object $X$ such that $JT/X = Sh(K)$ is actually the free JT algebra on one generator. Meaning that given an object $Z$ of JT one recover the corresponding JT-algebra by looking at $Hom(X,Z)$.
So what you are asking is what is $Hom(X,f^*s)$, Let me call $h$ the geometric morphisms $JT/X \to JT$ and $g:JT/X = Sh(K) \to Sets$ the unique geometric morphisms. Now, as $h$ is étale, it has an exceptional further left adjoint $h_!$ and $X= h_! 1$, so, $Hom(X,Z)$ can be more generally written as $Hom(h_! 1 ,Z ) = Hom(1,h^*Z)$ and hence $Hom(X,f^*s) = Hom(1,h^* f^* s) = Hom(1,g^* s) $
And $g^*s$ is the sheaf of locally constant functions with values in $s$ on $K$, so $Hom(1,g^*s)$ is the set of locally constant functions from $K$ to $s$.
So at the end of the day one has:
$$ f^* (s) = \{ \text{locally constant functions } K \to s\} $$
where $K=2^\mathbb{N}$ is the Cantor space. With some additional calculation (related to how $JT/X$ identifies with $Sh(K)$), one can show that the JT algebra structure on $f^*(s)$ is obtained from the homeomorphisms $K = K \coprod K$ obtained by separating along the  first component of $K =  2^\mathbb{N}$
