Corollary 5.59.
For every acyclic class \(L\), we have
\[L \;=\; L^{\mathrm{epi}} \,\vee\, L^{\mathrm{mono}},\]
i.e. \(L\) is the smallest acyclic class containing both \(L^{\mathrm{epi}}\) and \(L^{\mathrm{mono}}\).
Proof
It is clear that \(L\) contains both \(L^{\epi}\) and \(L^{\mono}\). Conversely, let \(L'\) be an acyclic class containing both parts. For every \(f\in L\), Lemma 5.52 places \(\coim(f)\) in \(L^{\epi}\) and \(\im(f)\) in \(L^{\mono}\). Hence \(f=\im(f)\coim(f)\) lies in \(L'\), proving \(L\subseteq L'\).