Homotopy groups of a spectrum

Definition 4.4.21. For a spectrum \(X\) and an integer \(k\), we define its \(k\)-th homotopy group \(\pi _k(X) \in \Ab \) as \[ \pi _k(X) := [\S [k],X] := \pi _0\Hom _{\Sp }(\S [k],X). \] The assignment \(X \mapsto \pi _k(X)\) defines a functor \(\pi _*\colon \Sp \to \Ab \).