Universal element in Brown's construction

Definition 3.4.4 (Universal element). Let \(n\geq 1\). An element \(\xi \in F(Z)\) is \(n\)-universal if \[ T_\xi \colon [S^k,Z]_*\longrightarrow F(S^k) \] is a bijection for \(1\leq k<n\) and a surjection for \(k=n\). It is universal if this map is a bijection for every \(k\geq 1\).