Does the direct sum have a universal property in the category of groups

Question

In other categories, like modules, the direct sum is the coproduct, giving it a neat category theory description. The direct product of groups, which is very similar to it, is the product in the category of groups. And when restricted to Abelian groups, the direct sum becomes a coproduct. The direct sum seems to be a very general and natural construction even for general groups, so it seems like it should have a category theoretic description.

So is there any universal property which defines the direct sum in the category of groups?

Answer 1

The (improperly named) "direct sum" of a family $(G_i)_{i\in I}$ of (non necessarily abelian) groups (I prefer to call it "restricted sum") is the subgroup $$\sum_{i\in I}G_i:=\left\{\left.g\in\prod_{i\in I}G_i~\right|~\{i\in I\mid g_i\ne 1_i\}\text{ is finite}\right\}.$$ Its universal property, involving the natural injective morphisms $\varphi_j:G_j\to\sum_{i\in I}G_i,$ is the following:

For every group $H$ and any family of morphisms $f_i:G_i\to H$ with commuting ranges, i.e. such that $$f_i(x)f_j(y)=f_j(y)f_i(x)$$ for every distinct indices $i,j\in I$ and every elements $x\in G_i,y\in G_j,$
there is a unique morphism $f:\sum_{i\in I}G_i\to H$ such that $$\forall i\in I\quad f\circ\varphi_i=f_i.$$