Lemma about additivity and some variation of the depolarising channel

First we prove Lemma about the additivity and the multiplicativity. This Lemma states that for a unitary covariant channel $\Psi$ and a channel $M$ having some property the additivity of the minimal output entropy of $\Psi$ implies that of $\Psi\circ M$, and the multiplicativity of the maximal output $p$-norm of $\Psi$, that of $\Psi\circ M$. Next we introduce a new class of channel for which the additivity of minimal output entropy and the additivity of the Holevo capacity hold.