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)

thank you.