Example 5.78.
Let \(L=\Conn_n\) be the \(n\)-connected maps. As a modality this is generated by the constant map \(S^{n+1}\to *\) in \(T\). The pushout product of \(S^{n+1} \to *\) with \(S^{m+1} \to *\) is \(S^{n+m+3} \to *\), so by Lemma 5.74 we get
\[(\Conn_n)(\Conn_m) \;=\; \Conn_{n+m+2}.\]
As a special case, for \(\EffEpi = (S^0 \to *)^{m}\) we get \(\Conn_n = \EffEpi^{\,n+2}\).
More generally, note that for any morphism \(u\colon X \to Y\), the pushout product \((S^0\to *) \ssquare u\) is the codiagonal \(\nabla_u\colon Y \sqcup_X Y \to Y\). It follows that for every acyclic class \(L\) we have
\[\EffEpi \cdot L \;=\; \nabla(L)^{m}.\]