 Universal cohomology class of an Eilenberg–MacLane space

For abelian $G$ and $n\ge 1$, the identity of $G$ determines this class under $H^n(K(G,n);G)\cong \mathrm{Hom}(G,G)$. Pulling it back realizes representability of cohomology by Eilenberg–MacLane spaces. A map inducing this class on a fiber induces the identity of its unique positive homotopy group.