Denote by Ab the category of abelian groups and group homomorphisms and by CAb the category of compact Hausdorff abelian groups and continuous group homomorphisms. It follows from Pontryagin duality that CAb is equivalent to Ab$^{op}$, where $^{op}$ denotes the opposite category. This is a consequence of the fact that a locally compact group is compact if and only if its dual is discrete and viceversa.
Now, Ab has both products and coproducts, so the same is true for Ab$^{op}$ and therefore also for CAb. Products in CAb are just the usual direct products endowed with the product topology. On the other hand, I find it much harder to understand coproducts in this category. Given a family $(G_i)_{i\in I}$ of compact abelian groups, one way to describe their coproduct is the following: $$\coprod_{i\in I}G_i=\widehat{\prod_{i\in I}\widehat G_i}$$ where the hat denotes the Pontryagin dual and the product is endowed with the discrete topology.
Is there a "more concrete" way to describe the coproduct in CAb, that does not make use of duals?