Proposition 6.105.
Let \(T\) be a topos, and suppose \(T_0 \subseteq T\) is a full subcategory of \(n\)-coherent objects such that every \(X \in T\) is covered by objects of \(T_0\). Then the following conditions hold:
A morphism \(f\colon Y \to X\) is relatively \(n\)-coherent if and only if for every map \(U \to X\) with \(U \in T_0\) the pullback \(U \times_X Y\) is \(n\)-coherent.
An object \(X \in T\) is \((n+1)\)-coherent if and only if \(X\) is quasi-compact and for every pair of maps \(U \to X \leftarrow V\) with \(U ,V \in T_0\), the pullback \(U \times_X V\) is \(n\)-coherent.