Proof.
By Remark 2.2.6, the braiding complexes are quasi-isomorphic to the associated permutation bimodules, concentrated in homological degree \(0\). As the permutation bimodules implement the symmetric braiding, this means that the braiding complexes constructed as (shifted) Rouquier complexes of the cabled crossings become the canonical symmetric braiding in \(h_1\mathrm{DMor}^{\mathrm{poly}, \mathrm{gr-perf}}(\mathrm{mod}_k^{\mathbb{Z}})\). ◻