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
