Simple ribbon fusion dimensions are nonzero

By [F1], $\mathrm{Hom}(\mathbf{1},i\otimes i^{\vee })$ is one-dimensional, so the simple unit occurs exactly once in the semisimple decomposition of $i\otimes i^{\vee }$. The coevaluation is nonzero, hence its image is that unique unit summand. The map ${\mathrm{ev}}_{i^{\vee }}(j_i\otimes 1_{i^{\vee }})$ is nonzero: $j_i$ is an isomorphism and evaluation is nonzero. As a map from a semisimple object to the simple unit, it can be nonzero only on a unit summand; because there is exactly one such summand, its restriction to the coevaluation image is nonzero.

The composite in [F3] is precisely that nonzero restriction followed by the unit inclusion, so it is a nonzero endomorphism of $\mathbf{1}$. Since $\mathrm{End}(\mathbf{1})=\mathbb{C}$, this endomorphism is the scalar $d_i\mathrm{\ne }0$.