Corollary 5.76.
Let \(K\) and \(L\) be monogenic acyclic classes. Then \(KL\) is monogenic. If \(K\) and \(L\) are of small generation, then so is \(KL\).
Proof
By monogenicity, we may write \(K=\Sigma^m\) and \(L={\Sigma'}^m\) with \(\Sigma=K\cap\Mono\) and \(\Sigma'=L\cap\Mono\). The class \(\Sigma\ssquare\Sigma'\) consists of monomorphisms, so Lemma 5.74 shows that \(KL\) is monogenic. If \(K\) and \(L\) are of small generation, Lemma 5.56 allows us to choose \(\Sigma\) and \(\Sigma'\) to be sets, which also proves small generation of \(KL\).