I would like to prove the following proposition:
Let $X$ and $Y$ be based topological spaces and let $[X,Y]$ be the set of homotopy classes of based maps $X\to Y$. If for every $X$, $[X,Y]$ is a group with unit the homotopy class of the constant map and that $f^*:[X',Y]\to[X,Y]$ is a homomorphism, where $f:X\to X'$, then $Y$ is an H-space.
My attempt is, let the product of $[Y,Y]$ be denoted by "$+$" (though it may not be commutative) and let $p_1$ and $p_2$ be the two natural projections from $Y\times Y$ to $Y$, then define $m$ and $i$ by $$[m]=[p_1]+[p_2]$$ and $$[i]=-[id_Y].$$
Then I can prove all conditions in the definition of H-group except that $$m(m\times id)\simeq m(id\times m):Y\times Y\times Y\to Y.$$
My attempt:
$[m(m\times id)]=(m\times id)^*([m])=(m\times id)^*([p_1]+[p_2])=(m\times id)^*([p_1])+(m\times id)^*([p_2])=[p_1(m\times id)]+[p_2(m\times id)],$
and similarly
$[m(id\times m)]=[p_1(id\times m)]+[p_2(id\times m)].$
But how to proceed? How can I use the associate property of $[Y\times Y,Y]$?