Universal property of Rep(SO(n, k)) with orientation and 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(SO(n, k)) is the free 2-rig on an object x equipped both with an isomorphism An(x) ≅ I and with a self-dual structure whose counit ε: x ⊗ x → I is symmetric.

Background

Special orthogonal groups impose both preservation of a symmetric bilinear form and determinant one. In the 2-rig setting, this translates to a symmetric self-duality together with a chosen trivialization of the top exterior power.

The conjecture posits that Rep(SO(n, k)) is the free 2-rig on an object carrying these two pieces of structure, extending the proposed universal properties for SL and O to SO.