I just come across a definition of the unitary fusion category:

A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have:

- We have a Hilbert space structure on each Hom space, and we denote the inner product by $\langle \cdot, \cdot \rangle$.
- We have a contravariant endofunctor * on $\mathcal{C}$ which is identity on objects.
- We have $\|fg\| \leq \|f\|\|g\|$ and $\|f^*f\| = \|f\|^2$ for each $f \in Hom(Y,Z)$ and $g \in Hom(X,Y)$.
- We have $(f \otimes g)^* = f^* \otimes g^*$ for any morphism $f$ and $g$.
- All structure isomorphisms for simple objects are unitary.

I am confused about the notation in condition 4. Does $\|f\| = \sqrt{ \langle f, g\rangle}$ or $\|f\|$ means the operator norm? From my understanding, I guess $\|f\|$ in the definition should means the operator norm. Would anybody please clarify this for me? Thank you!