My question is about the proof of Proposition A.6.1.6 in Lurie's Spectral Algebraic Geometry, which says the following:
Let $\mathcal{X}$ be any $\infty$-topos and denote by $\mathcal{X}^{coh}$ the full subcategory of the coherent objects. Then $\mathcal{X}^{coh}$ is a local $\infty$-pretopos.
In the proof, to show that $\mathcal{X}^{coh}$ is closed under geometric realizations of groupoid object, Lurie in a crucial way uses Proposition A.2.1.5 which states that if the pullback of a morphism along some effective epimorphism is relatively $n$-coherent, then the original morphism is already relatively $n$-coherent. This proposition, however, is only applicable if one assumes furthermore local $n$-coherence of $\mathcal{X}$, which again enters the proof in a very crucial way (no find an $n$-coherent cover of the object $U$ in the notation there).
Thus, it seems like the proof only works if $\mathcal{X}$ is already locally coherent, unless I just misunderstood the argument. Can it still somehow be salvaged for a general $\infty$-topos, as was claimed? I unfortunately also didn't find a proof of a classical analogon of this statement in the literature.
Best Answer
If $X_0\to X$ is an effective epimorphism and $X_0$ is locally $n$-coherent, then $X$ is also locally $n$-coherent: every $Y$ over $X$ is covered by $Y\times_XX_0$, which is in turn covered by a coproduct of $n$-coherent objects. So in the proof of A.6.1.6 we know beforehand that $X$ is locally $n$-coherent for all $n$, hence all the results that assume local $n$-coherence apply.