ScalingStacks

Proof. Every element of $R_{\xi_{2}}^{-}(S,T)$ is of the form $(\operatorname{id}\nolimits_{T}\boxtimes\zeta)\cdot(\alpha\boxtimes\operatorname{id}\nolimits_{-1})$ for some $\zeta$ admissible class of paths starting at $-1$ and $\alpha$ a braid starting at $S$.

Every element of $L_{\xi_{1}}^{+}(S,T)$ is of the form $([m_{i}^{+}\to 1]\boxtimes\operatorname{id}\nolimits_{T})\cdot\alpha$ for some braid $\alpha$ starting at $S$.

It follows that every element of $(R_{\xi_{2}}^{-}L_{\xi_{1}}^{+})^{n}$ is of the form

$$(\operatorname{id}\nolimits\boxtimes\zeta_{1})\wedge([m_{i}^{+}\to 1]\boxtimes\operatorname{id}\nolimits)\wedge\cdots\wedge(\operatorname{id}\nolimits\boxtimes\zeta_{n-1})\wedge([m_{i+n-2}^{+}\to 1]\boxtimes\operatorname{id}\nolimits)\wedge(\operatorname{id}\nolimits\boxtimes\zeta_{n})\wedge\alpha$$

for some $i\geq 0$ and $\zeta_{r}$ an admissible class of paths starting at $-1$ for $1\leq r\leq n$. The image by $\mu_{n}$ of such an element is

$$\bigl(([m_{i+r-1}^{+}\to r])_{1\leq r\leq n-1}\boxtimes\operatorname{id}\nolimits\bigr)\circ\alpha)\boxtimes\zeta_{1}\boxtimes(\zeta_{2}\circ[-2\to-1])\boxtimes\cdots\boxtimes(\zeta_{n}\circ[-n\to-1]).$$

It follows that $\mu_{n}$ is injective, hence it induces an isomorphism $(R_{\xi_{2}}^{-}L_{\xi_{1}}^{+})^{n}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}C_{n}$.

Let $L$ be the image of $\lambda\circ(T_{1}\otimes 1-1\otimes T_{1})$. We have $\nu_{2}=\mu_{2}\circ\lambda$. It follows that $\mu_{2}(L)=(T_{1}\otimes 1-1\otimes T_{1})(D_{2})$, since $D_{2}$ is the image of $\nu_{2}$ (Lemma 8.2.5). The theorem follows now from Corollary 8.2.16 and Theorem 8.2.1. ∎