# Geometric surjections are not stable under pullback

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!