ScalingStacks

Remark 8.2.6. Consider $\bar{\xi}_{1}^{+}:{\mathbf{R}}_{>0}\to Z^{{\operatorname{opp}\nolimits}},\ x\mapsto\xi_{2}^{-}(-x)$ and $\bar{\xi}_{2}^{-}:{\mathbf{R}}_{<0}\to Z^{{\operatorname{opp}\nolimits}},\ x\mapsto\xi_{1}^{+}(-x)$. There is an isomorphism $(Z^{\operatorname{opp}\nolimits})_{\bar{\xi}}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}(Z_{\xi})^{\operatorname{opp}\nolimits}$ that is the identity on $Z$ and $x\mapsto-x$ on ${\mathbf{R}}$. This provides an isomorphism $({\mathcal{S}}^{\bullet}(Z_{\xi}))^{\operatorname{opp}\nolimits}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}^{\bullet}(Z^{{\operatorname{opp}\nolimits}}_{\bar{\xi}})$. It induces isomorphisms

$$\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup(-n,-1),T\sqcup(1,n))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\operatorname{opp}\nolimits})}(T\sqcup(-n,-1),S\sqcup(1,n)).$$

This restricts to isomorphisms between $A_{n}$ (resp. $B_{n}$, $D_{n}$, $E_{n}$, $F_{n}$) for $Z$ and $A_{n}$ (resp. $B_{n}$, $D_{n}$, $F_{n}$, $E_{n}$) for $Z^{\operatorname{opp}\nolimits}$.