Proof of theorem 2.5.2.
theorem 2.5.4 indeed implies theorem 2.5.2. Namely it follows from Theorem 2.2.4 that the \(\beta_{m,n}\) are invertible and satisfy the hexagon axioms ([001L]) and thus form the components of a natural transformation \(\boxtimes\circ (h_1 K_{\mathrm{loc}}\times h_1 K_{\mathrm{loc}}) \Rightarrow \boxtimes^{\mathrm{op}}\circ (h_1 K_{\mathrm{loc}}\times h_1 K_{\mathrm{loc}})\) by Theorem 2.5.4. ◻