Lemma A.5.
Let \((L,R)\) be a factorization system on \(C\), and consider morphisms \(X \xrightarrow{f} Y \xrightarrow{g} Z\).
Assume \(C\) admits pullbacks. If \(gf \in R\) and \(\Delta_g \in R\), then \(f \in R\).
Assume \(C\) admits pushouts. If \(gf \in L\) and \(\nabla_f \in L\), then \(g \in L\).
Proof
We prove (1); the proof for (2) is dual. We may factor \(f\) as the composite
\[X \xrightarrow{(\id_X, f)} X \times_Z Y \xrightarrow{\pr_Y} Y.\]
The second map is a base change of \(gf\colon X \to Z\) along \(g\), hence lies in \(R\) because \(gf \in R\). The first map is a base change of the diagonal \(\Delta_g\colon Y \to Y \times_Z Y\) along the map \(f \times \id_Y \colon X \times_Z Y \to Y \times_Z Y\). Since \(\Delta_g \in R\), it follows that \((\id_X, f) \in R\). Since \(R\) is closed under composition, we conclude \(f \in R\).