ScalingStacks

Remark 8.1.6. Assume $\xi$ is terminal for $(Z,M)$. Consider $f:Z\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}Z$ an isomorphism of curves fixing $M$. Note that $f\circ\xi$ is terminal for $(Z,M)$ and the map $f$ induces an isomorphism $L_{\xi}^{\bullet}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L_{f\circ\xi}^{\bullet}$.

Consider now another injective morphism of curves $\xi^{\prime}:{\mathbf{R}}_{>0}\to Z$ such that $\xi^{\prime}$ is terminal for $(Z,M)$. Assume there is a connected open subset $U$ of $Z_{u}$ containing $\overline{\xi({\mathbf{R}}_{>0})}$ and $\overline{\xi^{\prime}({\mathbf{R}}_{>0})}$ and assume the canonical orientations on $\xi({\mathbf{R}}_{>0})$ and $\xi^{\prime}({\mathbf{R}}_{>0})$ extend to an orientation of $U$. There is an isomorphism of curves $f:Z\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}Z$ fixing $Z\setminus U$ such that $\xi^{\prime}=f\circ\xi$. It induces an isomorphism $L_{\xi}^{\bullet}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L_{\xi^{\prime}}^{\bullet}$, and that isomorphism does not depend on the choice of $f$.