Eilenberg–MacLane functor preserves colimits

Proof. Since it is exact, it remains to show it preserves arbitrary coproducts: the map \(\bigoplus _{i \in I} H(A_i) \to H(\bigoplus _{i \in I} A_i)\) is an isomorphism of spectra. This may be tested at the level of homotopy groups. Combining with Proposition 6.2.3, this then follows from the fact that both \(\pi _n(-)\colon \Sp \to \Ab \) and \(H_n(-)\colon \D (\Z ) \to \Ab \) preserve coproducts. □