Monogenic parts via images

The second relation follows from the first, since \(\Sigma^c = (\Sigma^{\Delta})^m\) by Theorem 5.39. The last two claims immediately follow by writing \(L = \Sigma^m\) and noting that \(\im(\Sigma)\) and \(\im(\Sigma^{\Delta})\) are small. We need to show the first relation.Let \(L := \Sigma^m\). By Lemma 5.52, we have \(\im(\Sigma) \subseteq L^{\mono}\). Since \(L^{\mono}\) is an acyclic class, it follows that \(\im(\Sigma)^m \subseteq L^{\mono}\). For the converse, consider the class \(L'\) of morphisms \(f\colon A \to B\) such that \(\im(f) \in \im(\Sigma)^{m}\). We are done if we can show that \(L \subseteq L'\), since then \(L \cap \Mono = \im(L) \subseteq \im(\Sigma)^{m}\) and thus also \(L^{\mono} = (L \cap \Mono)^m \subseteq \im(\Sigma)^{m}\). By construction we have \(\Sigma \subseteq L'\), so it remains to show that \(L'\) is an acyclic class.To this end, note that \(\im(\Sigma)^m = \im(\Sigma)^c\) by Corollary 5.43, so in particular \(\im(\Sigma)^m\) is a congruence of small generation. Consider now the quotient map \(T \to T/\im(\Sigma)^m\). This localization preserves effective epimorphisms because it preserves colimits, and it preserves monomorphisms because it preserves finite limits. It therefore preserves epi–mono factorizations. It follows that \(f \in L'\) if and only if the morphism \(f\) is sent to an effective epimorphism under this localization functor. Since the effective epimorphisms in \(T/\im(\Sigma)^c\) form an acyclic class, the same follows for \(L'\), finishing the proof.