Universal property of a free product
= Universal property of a free product
= Universal properties of a free product
{synonym}
Given <group homomorphisms> $f_A:A\to H$ and $f_B:B\to H$, there is exactly one <group homomorphism> $A*B\to H$ extending both. Its value on an alternating normal form is the product of the factor-map values. Thus the <free product> is the coproduct in the <category of groups>.