The Brauer group of a braided monoidal category $\mathcal{C}$ is defined in general in this paper. Essentially it's defined as the equivalence classes of Azumaya algebras in $\mathcal{C}$ (see the paper for the definition of an Azumaya algebra in a category) where two Azumaya algebras $A,B$ are called equivalent if there exist faithfully projective objects $P,Q$ such that $$A\#[P,P]\cong B\#[Q,Q],$$ here $[P,-]$ is the right adjoint functor to $-\otimes P$, i.e. $[P,-]$ is the internal Hom and one can view $[P,P]$ as a kind of endomorphism ring. The group operation is given by $\#$ where $A\#B$ is just the ordinary tensor product of two algebras and the multiplication on $A\#B$ is defined by using the braiding.
Let $H$ be a coquasi-triangular Hopf algebra and consider the braided monoidal category $^H\text{Mod}$ of $H$-comodules. One can consider the Brauer group $\text{Br}(^H\text{Mod})$. How much is known about these Brauer groups? Given a very specific $H$, what kind of techniques are there to compute this Brauer group? References and/or explanations are very welcome.
Also, I noticed that in many instances one considers other Brauer groups as well, i.e. those that look more restrictive algebras than Azumaya algebras or in some cases more general types of algebras. Are there other Brauer groups that are easier to handle somehow?