Direct proof of determinant bicharade coherence

Provide a proof of the commutativity of the determinant-based coherence diagram (27b) for fdVec-bicharades that avoids the use of standard-form decompositions of diagrams S → T → Z, thereby yielding a more direct and conceptual argument for the determinant bicharade’s coherence.

Background

In Example 31, the determinant (top exterior power) is shown to define a bicharade over the category of finite-dimensional vector spaces. The proof of the key coherence diagram (27b) proceeds by reducing any diagram S → T → Z to a particular ‘standard form’ and then applying determinant isomorphisms along exact sequences.

The authors express belief that a simpler, more conceptual proof exists which avoids reliance on standard forms, but they have been unable to find such an argument. Producing a direct proof would strengthen the conceptual understanding of determinants within the bicharadic framework.