On the definition of the direct sum of not necessarily abelian groups

Question

Consider the following definition for the direct sum of groups:
Definition. Let $(A_i)_{i\in I}$ be a family of groups. Then we define the direct sum of the family $(A_i)_{i\in I}$ as the subgroup of the direct product $\displaystyle \prod_{i\in I}A_i$ that contains all the elements $(a_i)$ such that there are only finitely many $i$ such that $a_i$ is not the identity element of $A_i$.
Does this definition make sense for families of groups that are not necessarily abelian (I know that this definition commonly applies to families of abelian groups)? My understanding is the following: this is going to be a subgroup of the direct product even if the groups are not abelian, but it will not retain the properties it has if all the groups are abelian. If we want to retain the properties of the direct sum of abelian groups, then we have to use the free product, which is a different construction from the one in my definition. So, am I right?

Answer 1

Yes, you are right. There is a notion of direct sums of (infinitely many) groups, see e.g. Wikipedia, though in my eyes it could be an ambiguous term, as one usually uses speaks about coproducts when people talk about direct sums.

You are also right that this definition does not satisfy the universal property of coproducts. As you noted, the coproduct in the category of groups $\mathbf{Grp}$ is given by the free product. Since coproducts (of the same components) are canonically isomorphic, surely your notion is not the coproduct in $\mathbf{Grp}$.

That is, the forgetful functor $\mathbf{Ab} \to \mathbf{Grp}$ is not cocontinuous (i.e. does not preserve small colimits) which in particular shows that it does not admit a right adjoint.