Universal property of Rep(O(n, k)) via a symmetric self-duality (algebraically closed k)

Establish that for any algebraically closed field k of characteristic zero and integer n ≥ 1, the 2-rig Rep(O(n, k)) is the free 2-rig on a self-dual object x of dimension n (i.e., An(x) invertible) whose counit ε: x ⊗ x → I is symmetric, equivalently satisfying ε ∘ Sx,x = ε.

Background

Orthogonal groups are characterized by preserving a nondegenerate symmetric bilinear form. Over algebraically closed fields, all such forms of a given dimension are isomorphic, which avoids signature issues and makes a universal property plausible.

The conjecture formalizes this by asserting that Rep(O(n, k)) is freely generated by a self-dual object with a symmetric counit, capturing the orthogonal form categorically.