Symmetric coherence

By [L1] and [L3], choose a braided monoidal equivalence from $\mathcal{C}$ to a strict braided monoidal category $\mathcal{D}$. Because the braiding of $\mathcal{C}$ is involutive, the transported braiding on $\mathcal{D}$ is also involutive, so $\mathcal{D}$ is strict symmetric.

In the strict symmetric category $\mathcal{D}$, every canonical map between two tensor words with the same ordered letters is built from adjacent swaps of neighboring factors. By [F1] and [L4], the resulting composite depends only on the permutation carrying the source word to the target word, not on the chosen decomposition of that permutation into adjacent transpositions.

Transport this canonical map back across the equivalence chosen in step 1.1. Faithfulness of an equivalence preserves uniqueness, so the resulting canonical map in $\mathcal{C}$ depends only on the same permutation. Hence any two canonical symmetric composites with the same source, target, and underlying permutation are equal.