# Can “premodular” be relaxed as a condition for uniqueness of Bruguieres/Mueger modularization?

Suppose that C is a ribbon monoidal category with dominant ribbon functors F_1: C->D_1 and F_2: C->D_2 such that D_1 and D_2 are modular tensor categories, does it follow that D_1 and D_2 are equivalent as MTCs? Here dominant means that every object in the target is a summand of an object in the image of the functor.

This is certainly true if C is premodular (semisimple with finitely many simple objects) as was proved by Bruguieres. What if C is not premodular? I haven't been able to locate a more general statement in the literature.

The particular case I have in mind is where C is the Kuperberg G_2-spider specialized to q a particular root of unity. After semisimplification C is in fact premodular, but actually proving that is likely to be a lot of work (it would require writing down inductive formulas for simples, etc.).

Let B be the Fibonacci category and let B' be its Galois conjugate, let x and x' be the nontrivial objects. Let D_2 be the Deligne tensor product of B and B'. Let F_2 be the functor sending V_2 to $x \boxtimes x'$, this exists by the universal property of Temperley-Lieb (i.e. $x \boxtimes x'$ is symmetrically self-dual and has quantum dimension -1). This functor is dominant because $F_2(V_2 \otimes V_2)$ has every object of D_2 as a summand.