The Khovanov-Seidel complexes give a weak derived braid action

Every $F_{\beta }$ is an equivalence. For $\beta \mathrm{\ne }1$, [L1] makes $G_{\beta }{\otimes }_{A_m}-$ an endofunctor of $C_m$, and the generator inverse relations of [L2], applied factor by factor with tensor associativity, identify its composites with the functor of the literal reversed inverse word as the identity; the empty-word functor is canonically the identity by [L6]. Thus $F_{\beta }$ is an equivalence with inverse the functor of that literal inverse word, up to the canonical unit identification. For $\beta =1$ the claim holds by the definition $F_1=\mathrm{Id}$.

The defining relations hold up to natural isomorphism. The inverse-cancellation relation is [L2], far commutativity is [L3], and the three-term Artin relation is [L4]; for each of these the two functors are respectively naturally isomorphic, and the isomorphisms are compatible with concatenation of words because both sides are computed by the same balanced tensor product of the word complexes [L1].

Well-definedness on braid words. Let $\sigma ={\tau }_1\cdots {\tau }_k$ be a word and let ${\sigma }^{\mathrm{\prime }}$ be obtained from it by one of the elementary moves of [L5]: inserting or deleting ${\sigma }_i^{\pm 1}{\sigma }_i^{\mp 1}$, commuting two far-apart letters, or replacing ${\sigma }_i{\sigma }_{i+1}{\sigma }_i$ by ${\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}$. Each move replaces $R_{\sigma }$ by a naturally isomorphic functor by step 1.2, since the tensor product identifies the segments of the word and the isomorphisms compose; by induction on the number of moves, any two words presenting the same element of $B_{m+1}$ yield naturally isomorphic functors. Hence the assignment $\beta \mapsto F_{\beta }$ is well defined up to natural isomorphism on the presented group, with $F_1$ represented by the identity functor rather than merely by an isomorphic empty-word functor.

The weak-action axioms. By definition $F_1={\mathrm{Id}}_{C_m}$. If $\beta ,\gamma \mathrm{\ne }1$, concatenating their chosen words gives $G_{\beta }{\otimes }_{A_m}{G}_{\gamma }\simeq G_{\beta \gamma }$ up to the natural isomorphism of step 2.1; tensoring with an input complex gives $F_{\beta }{F}_{\gamma }\cong F_{\beta \gamma }$, using [L6] when $\beta \gamma =1$. If one of $\beta ,\gamma$ is $1$, the corresponding composite is identified with the other functor by the canonical tensor unit isomorphism (and is literally composition with $\mathrm{Id}$ on the functor side). Thus the weak-action unit and pairwise-isomorphism conditions hold. No compositors satisfying a pentagon are produced or claimed.

Conclusion. The functors $F_{\beta }$ define a weak action of $B_{m+1}$ on $C_m$: the assignment is well defined up to natural isomorphism on words for the same braid (step 2.1), every value is an equivalence (step 1.1), and the identity functor is assigned exactly to $1$ with the pairwise isomorphisms supplied in step 3.1. No coherence upgrade is claimed.