Let B denote the braid groupoid, with objects being non-negative integers n∈Z≥0 and morphisms B(n,n)=Bn given by the braid group. Let C be a cocomplete rigid symmetric monoidal abelian category and F:B→C a monoidal functor. Moreover, let C:=∫b∈BF(b)∨⊗F(b) be the coend of F and V:=F(1).

Is there a “purely categorical construction” of a category CV , s.t. there is a braided monoidal equivalence CV≅CC 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 C generated by V.
- close it under direct sums
- …
- take the Karoubi envelope

**Answer**

**Attribution***Source : Link , Question Author : Bipolar Minds , Answer Author : Community*