Bihomomorphism of odd-order abelian groups functorially gives group generated by two abelian normal subgroups

Statement
Suppose $$A,B,C$$ are abelian groups and $$f: A \times B \to C$$ is a bihomomorphism, i.e., it is additive in both arguments. Then, the associated group generated by two abelian normal subgroups is defined as the group corresponding to an alternating bilinear map from one abelian group to another for the alternating bilinear map:

$$c: (A \times B) \times (A \times B) \to C$$

given by:

$$\! c((a,b),(a',b')) = f(a,b') - f(a',b)$$.