ScalingStacks

We show here that the previous isomorphism is functorial in $T\in{\mathcal{S}}_{M}(Z_{\xi})$. Consider the diagram

(8.3.1)

where $w=\left(\begin{matrix}w_{11}&w_{12}\\ 0&w_{22}\end{matrix}\right)$ (cf §5.4.2) with

$$w_{11}=(R_{\xi_{2}^{-}}(\mathrm{mult}\circ\Xi\operatorname{Hom}\nolimits))\circ(\tau L_{\xi_{1}^{+}}\operatorname{Hom}\nolimits)\circ(R_{\xi_{2}^{-}}\lambda\operatorname{Hom}\nolimits)$$

$$w_{12}=R_{\xi_{2}^{-}}\varepsilon\operatorname{Hom}\nolimits$$

$$w_{22}=(R_{\xi_{1}^{-}}(\mathrm{mult}\circ\Xi\operatorname{Hom}\nolimits))\circ(\sigma L_{\xi_{1}^{+}}\operatorname{Hom}\nolimits)\circ(R_{\xi_{2}^{-}}\rho\operatorname{Hom}\nolimits).$$