Let $\mathcal{C}$ be a $k$-linear abelian monoidal category and $(V,c)$ a braided object in $\mathcal{C}$. This means that $c \in Aut(V \otimes V)$ and satisfies the Yang-Baxter equation. Let $\mathcal{C}'_V$ be the groupoid with objects given by tensor powers $V^{\otimes n}$ and morphisms coming from the braid group action $B_n \to Aut(V^{\otimes n})$. Now we can close $\mathcal{C}'_V$ under direct sums and take the Karoubi envelope, giving a braided monoidal category $\mathcal{C}_V$. Obviously, $V$ is an object in $\mathcal{C}_V$ and we can take the Nichols algebra $\mathfrak{B}(V)$ in $\mathcal{C}_V$.
Is there an abstract characterization of the object $\mathfrak{B}(V)$ in $\mathcal{C}_V$?
I know the universal property of the Nichols algebra, but here we know more, since $\mathcal{C}_V$ is generated by $V$. I think that in some sense, $\mathfrak{B}(V)$ "contains all the information" of $\mathcal{C}_V$, but I'm not sure how exactly. Is it maybe a minimal generator?