I looked at my homotopy theory lecture notes and we had the following similar result: $X$ H-CoGroup, $Y$ $H$-Group, then both group structures defined of on [X,Y] agree. The proof goes roughly as follows: Call the upper products $\cdot$, resp. $*$. Inserting the definitions of thos those products, one can show , the following "distributivity"distributivity":
$(a\cdot b)*(c\cdot d)=(a * c)\cdot(b * d)$
Then one shows that both products have the same neutral element and finally
$f*g=(f\cdot 1) * (1\cdot g)=(f * 1)\cdot(1 * g)=f\cdot g$,
gives the result. That's the strategy of the proof in the case of $H$-(co-)groups.I don't know, whether the upper (weaker) assumptions also allow a proof like this.