ScalingStacks

Proof. Let $z=\zeta^{\prime}(0)=\zeta^{\prime\prime}(1)$. We have $\zeta^{\prime}\in I(\operatorname{id}\nolimits_{z},\theta_{\zeta(1)})$. Since $\bar{\zeta}^{\prime}=\theta_{\zeta(1)}\circ\zeta^{\prime}$ is smooth and has opposite orientation to $\zeta^{\prime}$, it follows that $\zeta^{\prime}(0+)\in\iota(C(z)^{+})$. Similarly, $\zeta^{\prime\prime}(1-)\in\iota(C(z)^{+})$. We deduce that $\zeta^{\prime}$ and $\zeta^{\prime\prime}$ have opposite orientations. ∎