I have a question about the proof of proposition $ 3.3.6(3)$ in “Tensor Categories” by Etingof et al..

This part states that for $ A$ , transitive unital $ \mathbb Z_+$ -ring, there is a unique character taking non-negative values on the basis elements.

The proof uses the fact that if $ \chi$ is a character, and $ f$ is a vector with entries $ \chi(Y)$ for basis elements $ Y$ , then $ f$ is an eigenvector of the matrix of left multiplication by $ \sum_{X\in I}X$ , where $ I$ is a set of basis elements.

But I don’t see why this holds. As I understand we want an equation: $ $ \sum_{X,Y\in I} \chi(Y) XY = \lambda \sum_{Z\in I} \chi(Z) Z , $ $ to hold, so $ \sum_{X,Y\in I} \chi(Y) c_{XY}^Z = \lambda \chi(Z)$ . But I do not understand what to do next.

In bibliographical notes there is a link to the paper https://arxiv.org/pdf/math/0203060.pdf (Lemma 8.3). Which uses more or less the same fact with $ X$ instead of $ \sum_{X\in I} X$ as a matrix, which is even more mysterious to me.

Can someone please clarify this for me?