What is an example of a functor $$F : \mathcal{C} \to \mathcal{D}$$ between two Grothendieck toposes which preserves colimits and finite products, but is not left exact (i.e., does not preserve pullbacks)?

I just assume that there is such an example, since otherwise the notion of an algebraic morphism between toposes would probably not include left-exactness. But I am not experienced enough in topos theory to see such an example.

My question is essentially why the forgetful strict $2$-functor from (toposes with algebraic morphisms) to (cocomplete symmetric monoidal categories with cocontinuous symmetric monoidal functors) is not fully faithful.


It has been answered on mathoverflow.

[I want to mark answered questions...]

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