Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^*$, we know that the decomposition of $V \otimes V^*$ must contain a copy of the trivial module. Is it true that it will only ever contain a single copy of the trivial representation?

Does this generalise to the setting of suitably qualified ** semisimple rigid monodial categories**? A specific guess:

*For a semisimple rigid braided monoidal category $\mathcal{C}$, with $V$ a simple object in $\mathcal{C}$, and $V^*$ its dual, the decomposition
of $V \otimes V^*$ into simple objects contains one and only one copy of the unit object of $\mathcal{C}$.*