Lemma 5.52.
Let \(L\) be an acyclic class, and let \(f\colon A \to B\) be a morphism with epi–mono factorization
\[A \overset{\coim(f)}{\twoheadrightarrow} \Im(f) \xhookrightarrow{\im(f)} B.\]
Then \(f\) is contained in \(L\) if and only if both \(\im(f)\) and \(\coim(f)\) are contained in \(L\).
Proof
If both \(\coim(f)\) and \(\im(f)\) lie in \(L\), then so does their composite \(f\). Conversely, assume \(f\in L\). Since \(L\) is local by Lemma 5.4 and \(\coim(f)\) is an effective epimorphism, we may test whether \(\coim(f)\) lies in \(L\) after base change along itself. The resulting projection \(A\times_{\Im(f)}A\to A\) is isomorphic to \(A\times_BA\to A\), because \(\Im(f)\to B\) is a monomorphism. This projection is a base change of \(f\) and hence lies in \(L\). Thus \(\coim(f)\in L\), and right cancellation applied to \(f=\im(f)\coim(f)\) gives \(\im(f)\in L\).