Lemma 2.40.
Given a commutative triangle
if \(gf\) is an effective epimorphism then so is \(g\).
Proof
We may factor the map \(g\) as
\[Y \xhookrightarrow{i_Y} Y \sqcup_{X} Z \xrightarrow{\lra{g,\id_Z}} Z,\]
where the first map is the inclusion of the \(Y\)-component and the second map is induced by the maps \(g\colon Y \to Z\) and \(\id_Z \colon Z \to Z\). The first map is an effective epimorphism, since it is a cobase change of the effective epimorphism \(gf\colon X \to Z\). The second map is an effective epimorphism by Lemma 2.38 since the canonical map \(Z \to Y \sqcup_X Z\) is a section. Their composite is \(g\), so \(g\) is an effective epimorphism by Lemma 2.37.