Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières modularisation/deequivariantisation exists.


*Is the resulting modular fusion category unitary? Is the modularisation functor unitary (dagger)?*


It feels to me like it should be obviously true, coming from the philosophy that the modularisation is a kind of generalised fibre functor. But I can't find a reference, nor can I write down the dagger structure.



Is the modularisation of a unitary fusion category always unitary?
The answer is yes to both questions (see [Müger's](http://arxiv.org/abs/math/9812040) paper). Müger's version of modularization is done in the unitary setting. The only new information that you need for full generality is that the unitary Drinfel'd center of a unitary fusion category is equal to the usual Drinfel'd center, (see [[M]](http://arxiv.org/abs/math/0111205)) 


More generally, a de-equivariantization of a unitary fusion category is again unitary (modularization corresponds to a specific kind of de-equivariantization). In fact, a $G$-de-equivariantization of a fusion category $\mathcal{C}$ is associated to a braided inclusion of Rep$(G)$ in $\mathcal{Z}(\mathcal{C})$. The de-equivariantization is the fusion category of left $\mathcal{O}\_k(G)$-modules in $\mathcal{C}$. Now, if $\mathcal{C}$ is unitary, then the Drinfel'd center is a unitary modular category. Therefore, $\mathcal{O}\_k(G)$ is a $Q$-system in $\mathcal{C}$. Thus, the de-equivariantization is unitary.


