The epigenic part of a modality

Let \(R := L^{\perp}\) and define \(R^{\epi} := (L^{\epi})^{\perp}\). We need to show that the pair \((L^{\epi},R^{\epi})\) is a factorization system. Orthogonality holds by construction, so it remains to check that any morphism \(f\colon A \to B\) admits an \((L^{\epi},R^{\epi})\)-factorization. Since \(L^{\epi} \subset L\), we get \(R \subseteq R^{\epi}\). Similarly, from \(L^{\epi} \subseteq \EffEpi\) it follows that \(\Mono \subseteq R^{\epi}\). Consider the \((L,R)\)-factorization \(A \xrightarrow{l} C \xrightarrow{r} B\) of \(f\), and consider the epi-mono factorization \(A \overset{\coim(l)}{\twoheadrightarrow} \Im(l) \xhookrightarrow{\im(l)} C\) of \(l\). By Lemma 5.52 we have that \(\coim(l) \in L \cap \EffEpi = L^{\epi}\), while \(r \circ \im(l) \in R \circ \Mono \subseteq R^{\epi}\) because the right class \(R^{\epi}\) is closed under composition.