Corollary 6.106.
A topos \(T\) is locally coherent if and only if there exists a full subcategory \(T_0 \subseteq T\) covering \(T\) which is closed under fiber products and such that every \(X \in T_0\) is quasi-compact.
Proof
The “only if” is clear: if \(T\) is locally coherent, we may take \(T_0 := T^{\mathrm{coh}}\) the subcategory of coherent objects, which by design are closed under fiber products.Conversely, we inductively use the proposition to show that \(T_0 \subseteq T^{n\text{-coh}}\) for all \(n\). In particular, \(T_0\) consists of coherent objects. Since they cover \(T\), we see that \(T\) is locally coherent by definition.