Minimal order semihypergroups of type U on the right, II

We prove a new characterization of the heart W_H and of the derived D_H of a finite semihypergroup H of type U on the right, in terms of its hyperproducts. Consequently we characterize the minimal size of the proper semihypergroups H of type U on the right such that W_H=D_H\not= H, W_H\not=D_H\not=H or W_H\not=D_H=H. Moreover, we obtain new results on the fundamental relations \beta and \gamma on finite semihypergroups of type U on the rght. In particular, we prove that \gamma is transitive, \beta is transitive if |H|\leq 8, and that there exist semihypergroups of type U on the right of size \geq 9 where \beta is not transitive. Hence, the minimal size for a semihypergroup of type U on the right to have a nontransitive \beta is 9.