"The coproduct in the category of sets is simply the disjoint union with the maps ij being the inclusion maps. Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. For example, the coproduct in the category of groups, called the free product, is quite complicated."
I am wondering, is it correct to say that the free product of groups is the group generated by the disjoint union of the summands with no relation? Therefore it is similar to the coproduct in the category of sets, where the coproduct is the set generated by the disjoint unions of the summands (the disjoint union itself). And in the category of abelian groups, is it correct to say the coproduct (the weak direct sum) is the abelian group generated by the disjoint union of the groups?