Generated modalities as saturations of base changes

Write \(\Sigma' := (\Sigma^{bc})^s\), the saturation of \(\Sigma^{bc}\). It is clear that \(\Sigma' \subseteq \Sigma^m\), so it remains to show that also \(\Sigma^m \subseteq \Sigma'\). To this end, consider the following class of morphisms We have \(\Sigma'' \subseteq \Sigma'\), so it remains to show that in fact \(\Sigma^m \subseteq \Sigma''\). We will do this by showing that \(\Sigma''\) is a saturated class closed under base change which contains \(\Sigma\).It is clear that \(\Sigma''\) is closed under base change. We claim that it is also saturated. It is clear that \(\Sigma''\) is closed under composition and contains all isomorphisms, so it remains to show that it is closed under colimits. Consider a diagram of maps \(X_{\bullet} \to Y_{\bullet}\) in \(\Sigma''\), and let \(Z \to \colim_i Y_i\) be a map. We need to show that the base change \(Z \times_{\colim_i Y_i} \colim_i X_i \to Z\) lies in \(\Sigma'\). By descent, we may write \(Z\) as \(\colim_i Z_i\), and hence this base change map may be written as the colimit of the maps \(Z_i \times_{Y_i} X_i \to Z_i\). Since each of these base changes lies in \(\Sigma'\) by definition of \(\Sigma''\), the claim follows from the fact that \(\Sigma'\) is closed under colimits.It remains to show that \(\Sigma''\) contains \(\Sigma\). In other words, given a morphism \(u\colon A \to B\) in \(\Sigma\) and an arbitrary map \(Z \to B\), we need to show that the base change \(Z \times_B A \to Z\) lies in \(\Sigma'\). To this end, let \(\Ee \subseteq T_{/B}\) be the full subcategory spanned by those \(Z \to B\) for which this condition holds. By definition of \(\Sigma'\), we have \(\Sigma^{bc} \subseteq \Sigma'\), and in particular \(\Ee\) contains all maps \(Z \to B\) with \(Z \in \Gg\). Since \(\Sigma'\) is closed under colimits, it follows from descent that \(\Ee \subseteq T_{/B}\) is closed under colimits. But since \(\Gg\) is a set of generators, it follows that \(\Ee = T_{/B}\), showing the claim.