In Lang’s Algebra, he constructs the coproduct of a family of groups in a similar way to the construction of the free group of a set $S$, $F(S)$. He later constructs the free product of $n$ groups and shows that if they are cyclic, then the free product is isomorphic to the free group on $n$ generators. I have also read elsewhere that the free product(of finitely many groups) is the coproduct of those groups. In light of the similar construction between the coproduct in the category of groups and $F(S)$, I am wondering if the coproduct of an arbitrary family {$G_i$} is also a free group. If you look in Lang, it seems that the coproduct of groups may be the free group generated by the set $S$ which has the same cardinality as the disjoint union of the groups {$G_i$}.
My apologies if this is trivial, I just can’t quite grasp whether or not it is true.