Summary: In order to avoid combinatorial rule explosion in fuzzy reasoning, the authors investigated in Part I [“Distributivity equations of implications based on continuous triangular norms. I“, IEEE Trans. Fuzzy Syst. 21, 153–167 (2012)] the distributivity equation of implication \(I(x,T_1(y,z))=T_2(I(x,y),I(x,z))\), when \(T_1\) is a continuous but not Archimedean triangular norm, \(T_2\) is a continuous and Archimedean triangular norm and \(I\) is an unknown function. In fact, it partially answered the open problem suggested by M. Baczyński and B. Jayaram in [“On the distributivity of fuzzy implications over nilpotent or strict triangular conorms“, IEEE Trans. Fuzzy Syst. 17, No. 3, 590–603 (2009)]. In this work we continue to explore the distributivity equation of implication \(I(x,S_1(y,z))=S_2(I(x,y),I(x,z))\), when both \(S_1\) and \(S_2\) are continuous but not Archimedean triangular conorms, and \(I\) is an unknown function. Here it should be pointed out that these results make difference with recent ones obtained in Part I [loc. cit.]. Moreover, our method can still apply to the three other functional equations related closely to this equation. It is in this sense that we have completely solved the open problem commented above.