\displaystyle \sum_{mn=d_1 d_2}f(d_1)g(d_2)=\\
=\sum_{mn=d_1 d_2}f((d_1,m)(d_1,n))g((d_2,m)(d_2,n))=\\
=\sum_{mn=d_1 d_2}f((d_1,m))f((d_1,n))g((d_2,m))g((d_2,n))=\\
=\sum_{mn=d_1 d_2}f((d_1,m))g((d_2,m))f((d_1,n))g((d_2,n))=\\
=\sum_{m=d_1 d_2}f(d_1)g(d_2)\sum_{n=d_1 d_2}f(d_1)g(d_2)=\\
=(f\star g)(m)(f\star g)(n)