# Categorical construction of comodule category of FRT algebra

Let $$\mathcal{B}$$ denote the braid groupoid, with objects being non-negative integers $$n \in \mathbb{Z}_{\geq 0}$$ and morphisms $$\mathcal{B}(n,n)=B_{n}$$ given by the braid group. Let $$\mathcal{C}$$ be a cocomplete rigid symmetric monoidal abelian category and $$F:\mathcal{B} \to \mathcal{C}$$ a monoidal functor. Moreover, let $$C:= \int^{b \in \mathcal{B}}\, F(b)^\vee \otimes F(b)$$ be the coend of $$F$$ and $$V:=F(1)$$.

Is there a "purely categorical construction" of a category $$\mathcal{C}_V$$ , s.t. there is a braided monoidal equivalence $$\mathcal{C}_V \cong \mathcal{C}^C$$ to the comodule category of the coalgebra $$C$$ (which is actually a coquasitriangular bialgebra).

Here, by "purely categorical construction" I mean (up to order and missing intermediate steps) something along the lines of

• take free monoidal subcategory of $$\mathcal{C}$$ generated by $$V$$.
• close it under direct sums
• ...
• take the Karoubi envelope