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Let $f:\mathcal{T} \rightarrow \mathcal{S}$ be a geometric morphism between two toposes. Let $g:\mathcal{L}\rightarrow \mathcal{S}$ be an open surjection (of toposes) and assume that the map $\mathcal{T} \times_{\mathcal{S}} \mathcal{L} \rightarrow \mathcal{L}$ turn $\mathcal{T} \times_{\mathcal{S}} \mathcal{L}$ into a coherent $\mathcal{L}$-topos.

Does $\mathcal{T}$ is a coherent $\mathcal{S}$-topos ?

I assume that all the toposes involved are Grothendieck toposes, or at least that the geometric morphisms are bounded.

Also does it work under different assumptions on the map $g:\mathcal{L}\rightarrow \mathcal{S}$ ? (like if it is a Proper surjection, or an Hyperconnected map)

Note: By cohenrent toposes, I mean as defined in section D3.3 of the elephant. Theorem C5.1.7 of the elephant cover a large number of similar properties for other type of toposes but does not mention Coherent toposes.

There is in the work of Moerdijk and Vermeulen (Here and Here) a few things about relative coherent morphisms which goes in this direction, but they do not talk about this kind of descent properties...

thank you.

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