Let’s just do the version without the sum from the paper. So you want: $$\sum_{Y\in I} \chi(Y) c_{XY}^Z = \lambda \chi(Z)$$

Just take $\lambda = \chi(X)$.

Hrm, I can’t seem to get it to work without assuming that the ring is based...

Let’s just do the version without the sum from the paper. So you want: $$\sum_{Y\in I} \chi(Y^*) c_{Z X^*}^{Y^*} = \sum_{Y\in I} \chi(Y) c_{XY}^Z = \lambda \chi(Z)$$

Now take $\lambda = \chi(X)=\chi(X^*)$.