I am reading Uniqueness of Moore Space $M(G,n)$, $n>1$ on Hatcher, but I have trouble understanding one sentence below "...there is a map $f:X\to Y$ inducing an isomorphism on $\pi_n$." It is unclean even why $\pi_n(X)\cong\pi_n(Y)$ holds.
If the space is simply-connected, then one can invoke Hurewicz's theorem so that $H_n=\pi_n$, so everything is done. But I don't think we can assume that.
Thanks in advance if someone can help!
