Woronowicz's Tannaka-Krein duality and free orthogonal quantum groups

Given a finite dimensional Hilbert space H and a collection of operators between its tensor powers satisfying certain properties, we give a category-free proof of the existence of a compact quantum group G with a fundamental representation U on H such that the intertwiners between the tensor powers of U coincide with the given collection of operators. We then explain how the general version of Woronowicz’s Tannaka-Krein duality can be deduced from this.