Exercise 16.10. on pg.356 in "Modern Classical Homotopy Theory."

Question

The question is asking to show that it suffices to prove Theorem 16.2 for a path connected space $X \in \mathcal{T_*}.$

Here is Theorem 16.2:

I believe that the answer of this question is similar to the answer of part(a) in the following link Show that for Abelian groups $G$ and $H, \bigl[K(G, n), K(H, n)\bigr] \cong \operatorname{Hom}(G, H).$ am I correct? if so could you please show me how? if not could you please show me the correct solution?