Lemma 5.60.
If \(L\) is a monogenic acyclic class, then \(L\) is a congruence. In particular, \(L^{\mono}\) is a congruence for every acyclic class \(L\).
Proof
By assumption we have \(L = L^{\mono} = (L \cap \Mono)^m\). Since \(L \cap \Mono\) consists of monomorphisms, it follows from Corollary 5.43 that \((L \cap \Mono)^m\) agrees with \((L \cap \Mono)^c\), hence is a congruence.