A unitary fusion category is a fusion category with a $C^*$-tensor structure. Hence, in principle, a fusion category could have more than one unitary structure. Does exist a fusion category with more than one (non equivalent) unitary structures.

Unitarity is equivalent to the existence of basis such that the $F$-matrices are unitary. This property is to be unitarizible. The question in matricial terms is: suppose that you have two different basis such that the $F$-matrices are unitary, Can be unitary gauged one basis in the other? or in oher words, there exist always unitary change of basis from one basis to the other basis?

[added by A. Henriques]

I see in the comments that there is some confusion in the interpretation of the question, so let me attempt a reformulation:

Is every equivalence $F:\mathcal C\to \mathcal D$ between unitary fusion categories naturally equivalent to a unitary equivalence (i.e., one which is a dagger functor)

I see that Galindo-Hong-Rowell have some results in that direction (see $\S$5.5, on page 21 of his paper), but they don't exactly addresses this question.

1more comment