ScalingStacks

Proof. Given $\gamma$ an oriented path in $Z$, the path $f(\gamma)$ is oriented. It is smooth if and only $f(\gamma)$ is smooth. This shows that if $\zeta$ is a smooth (resp. admissible) homotopy class of paths in $Z$, then $f(\zeta)$ is smooth (resp. admissible).

Consider now $\zeta$ a homotopy class of paths in $Z$ such that $f(\zeta)$ is smooth. Given $\gamma$ a minimal path in $\zeta$, then $f(\gamma)$ is minimal (Lemma 7.1.20). Since $f(\zeta)$ is smooth, it follows that $f(\gamma)$ is smooth (Properties 7.3.5(4)), hence $\gamma$ is smooth and finally $\zeta$ is smooth. ∎