An object of a braided category carries canonical braid actions

The strict case. For $n\ge 2$, assume first that $\mathcal{C}$ is strict. By [L1], $R:=c_{X,X}$ is a Yang–Baxter operator on $X$, so [L2] produces the homomorphism ${\rho }_n\text{\hspace{0.17em}⁣}:B_n\to \mathrm{Aut}(X^{\otimes n})$ with ${\rho }_n({\sigma }_i)=1_X^{\otimes (i-1)}\otimes R\otimes 1_X^{\otimes (n-i-1)}=1_X^{\otimes (i-1)}\otimes c_{X,X}\otimes 1_X^{\otimes (n-i-1)}$, and it is compatible with the inclusions. This gives the corollary in the strict case, together with the uniqueness of ${\rho }_n$.

The general case. For $n\ge 2$ in a general braided monoidal category, define ${\check{c}}_i$ on the fixed left-nested tensor power $X^{\otimes n}$ by conjugating the strict-model local braiding with the canonical associativity isomorphisms, as in [L3]. Each ${\check{c}}_i^{\pm 1}$ is a canonical composite built from associators, unitors and braidings, so [L3] shows that the composite does not depend on the chosen canonical isomorphisms and that the braid relations and the distant-commutativity relations between the ${\check{c}}_i$ hold, because the underlying braids of the two sides agree. The assignment ${\sigma }_i\mapsto {\check{c}}_i$ therefore satisfies the Artin relators of $B_n$ [L4], and [L4] gives a unique homomorphism ${\rho }_n\text{\hspace{0.17em}⁣}:B_n\to \mathrm{Aut}(X^{\otimes n})$ with these values; it is canonical because each ${\check{c}}_i$ is independent of the choices.

Zero and one strand. By [L4], $B_0$ and $B_1$ are trivial, so their unique group actions send $e$ to the identity. For $n=1$ both sides of the compatibility formula are $1_{X\otimes X}$. For $n=0$, tensor functoriality gives ${\rho }_0(e)\otimes 1_X=1_{\mathbf{1}\otimes X}$; conjugating by ${\lambda }_X$ gives $1_X={\rho }_1({\iota }_0(e))$, the stated compatibility.

Compatibility with the inclusions. For $n\ge 2$, in the strict model the compatibility is step 1.1. In general, both $\beta \mapsto {\rho }_{n+1}({\iota }_n(\beta ))$ and $\beta \mapsto {\rho }_n(\beta )\otimes 1_X$ are canonical composite assignments on braid words with the same underlying braid ${\iota }_n(\beta )$ in $B_{n+1}$; by [L3] they agree on every word, hence on every $\beta$ by [L4]. Thus ${\rho }_{n+1}({\iota }_n(\beta ))={\rho }_n(\beta )\otimes 1_X$.

Conclusion. Steps 1.1--1.2 construct the canonical homomorphisms in the strict and general case, and steps 1.3 and 2.1 give the compatibility with the standard inclusions. The construction uses only the braiding, its coherence and von Dyck's theorem; no choice principle is used, since all composites are finite and the canonical isomorphisms are explicitly determined.