A proof of Higgins' conjecture

Let f: G=* G(i) -> B=* B(i) be a group homomorphism between free products of groups. Suppose that G(i)f=B(i) of all i. Let H be a subgroup of G such that Hf=B. Then H decomposes into a free product H=*H(i) with H(i)f=B(i). Furthermore, H(i) decomposes into a free product of a free group and the intersection of H(i) with some conjugate of G(i). Higgins conjectured this in 1971 and now we prove it.