In the article A free group functor for stable homotopy theory, Barratt and Eccles define for each $X\in\mathsf{sSet}_{\ast}$, the free simplicial monoid $\Gamma^{+}X$.
Proposition 6.2 states
if $X$ is a $(n-1)$-connected pointed Kan complex, then $H_{i}(\Gamma^{+}X,X)=0$ for $i<2n$
My doubt starts when it is defined the quotient $\mathscr{D}=\Gamma^{+}_{p}X/\Gamma^{+}_{p-1}X$ and the following is asserted:
if $X$ is a $(n-1)$-connected, $\mathscr{D}_{p}X$ is clearly $(np-1)$-connected.
Actually, it is not clear for me. Could anybody help me to conclude this last assertion? thanks in advance.
