How can I prove that if for all $X \in \Bbb{Top_*}$ , $[X,W]_*$ has a natural group structure, then $W$ is an $H$-group?

Question

I know that if $\enspace[X,W]_*$ has a natural group structure, in particular $\enspace[W \times W, W]_*$ has it. If $p_1,p_2:W \times W \to W$ are the projections, it seems that defining $ \mu : W \times W \to W$ as the mapping satisfying $$ [p_1][p_2]=[\mu] $$ gives the $H$-multiplication on $W$, but I can't find the argument for proving that indeed $\mu$ satisfy the $H$-group axioms.

I appreciate any hints or suggestions.