How can I prove that central product of by is isomorphic to central product of by , i.e. .

(If G is a group and H,k be its subgroups, then G is central product of H by K if we have: 1)G=HK. 2)hk=kh for all h in H and all k in K, then we write G= .

Hint: We have where and is a homomorphism for which we have . Then . )