There are different questions about the coproduct in **Ab**, the coproduct in **Grp** and why they are so differents. However, I wonder why the following constructions cannot be extend for arbitrary groups:

Given two abelian groups $(A,+),(B,+),$ the set $A\times B$ endowed with the natural law $(a,b)+(a',b')=(a+a',b+b')$ together with the natural inclusions $\iota\_A:A\rightarrow A\times B$ and $\iota\_B:B\rightarrow B$ is the coproduct (for $A$ and $B$).

I guess I am using commutativity at some point (maybe to prove it satisfies the required universal property) but I cannot find where. If this construction were valid for any group in **Grp** then products and coproducts would match in **Grp** (for finite families), which is false. So, where is my mistake?

Thanks in advance