It is mentioned at the end of the introduction to Johnstone's Factorization systems for geometric morphisms, I that the pullback of a geometric surjection need not be a surjection, and hence that the surjection-inclusion factorisation system is not stable under pullback.
Is there a concrete counter-example out there for this statement?
Let $L$ be any non-trivial locale with no points; we can take the 'locale of surjections $\mathbb{N} \to A$, for $A$ an uncountable set' found in Example C1.2.8 of Johnstone's Sketches of an Elephant, for example. Then the presheaf category $[\mathcal{O}(L)^{\mathrm{op}},\mathbf{Set}]$ has enough points, so we have a geometric surjection to it from $\mathbf{Set}/K$ for some set $K$. Since geometric inclusion are stable under pullback, and all subtoposes of $\mathbf{Set}/K$ are of the form $\mathbf{Set}/K'$ for some $K' \subseteq K$, we find that: $$\require{AMScd}$$ \begin{CD} 0 @>>> \mathrm{Sh}(L);\\ @VVV @VVV \\ \mathbf{Set}/K @>>> [\mathcal{O}(L)^{\mathrm{op}},\mathbf{Set}], \end{CD} where $0$ is the degenerate topos, is a pullback diagram, and the upper morphism is certainly not a geometric surjection. So in a way, the fact that surjections are not stable under pullback arises from the same quirk that allows frames to not have enough points!