Do coherent toposes descend along open surjection?

Question

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

Does $\mathcal{L}$ 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 $g:\mathcal{T}\rightarrow \mathcal{S}$ ? (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 "descent" 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...

Answer 1

I recently found an answer for coherent locales, which was enough for my purpose although the argument can probably be expanded to an arbitrary coherent geometric morphism (see at the end, if someone can be bring some precision about this it would be very interesting for me).

Let $f : \mathcal{T} \rightarrow \mathcal{S}$ be an open surjection.

Let $\mathcal{L} \rightarrow \mathcal{S} $ be a locale in $\mathcal{S}$ such that, internally in $\mathcal{T}$, $f^* \mathcal{L}$ is coherent.

We will prove that $\mathcal{L}$ is already coherent in $\mathcal{S}$.

Internally in $\mathcal{T}$, $f^* \mathcal{L}$ is then the stone spectrum of a distributive lattice $I$, and $I$ can be chosen canonically: it is the set of open compact subspaces of $f^* \mathcal{L}$. In particular, when $\gamma: \mathcal{M} \rightarrow \mathcal{N}$ is an isomorphism between two coherent locales it induces an isomorphism between the corresponding distributive lattice, and hence the descent data on $f^* \mathcal{L}$ gives rise to a descent data on the corresponding distributive lattice $I$. (We are also using the fact that the pullback of a spectrum of a distributive lattice is the spectrum of the pullback of the distributive lattice)

Open surjections are of effective descent for sets (objects) hence $I$ (and its descent data) are of the form $f^* I'$ for some $I'$ a distributive lattice (in the base topos). The construction $I \mapsto \text{spec } I$ is compatible with pullback along geometric morphisms hence the isomorphism between $f^* \mathcal{L}$ and $\text{spec } I = f^* \text{spec } I'$ (which is compatible with the descent data simply because we used the isomorphism to construct them) descend into an isomorphism between $\mathcal{L}$ and $\text{spec } I'$ over $\mathcal{S}$, proving that $\mathcal{L}$ is coherent.

So one has the answer for coherent locales, and the open surjection can be replaced by any geometric morphism which is of effective descent for locales (claim: a geometric morphism which is of effective descent for locales is also of effective descent for objects).

This technique can maybe be extended to coherent toposes, but one needs an argument of descent for categories (and for "week descent data") because the distributive lattice of compact open subspace will be replaced by the pre-topos of coherent objects. I have never seen such a result of descent for categories stated but it will probably be not too difficult to obtain at least for hyperconnected geometric morphism.