Colimits of topoi along étale morphisms

We apply Proposition 4.60 to the wide subcategory \(\Topos^{\et}\subseteq \Topos\).First fix a topos \(S\). We claim that \((\Topos^{\et})_{/S}\subseteq \Topos_{/S}\) is closed under colimits. Under the equivalence from Proposition 4.38, a diagram of étale morphisms over \(S\) corresponds to a diagram \(X_{\bullet}\) in \(S\). If \(X=\colim_i X_i\), descent in \(S\) gives which says that the corresponding colimit in \(\Topos_{/S}\) is again étale over \(S\).It remains to verify the structure-map condition. Let \(T_{\bullet}\colon I\to \Topos^{\et}\) be a diagram, and let \(T=\colim_i T_i\) be its colimit in \(\Topos\). Passing to logoi, this is the limit of the opposite diagram in \(\Logos^{\et}\). By Lemma 4.59, each projection \(T\to T_i\) on the logos side is an étale morphism. Equivalently, each structure morphism \(T_i\to T\) on the topos side is étale.The criterion therefore shows that \(\Topos^{\et}\) is closed under colimits in \(\Topos\), and that the colimit computed in \(\Topos\) has the universal property in \(\Topos^{\et}\). Hence the inclusion preserves these colimits.