Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 3 / d

Let $b^{\prime }\in B$. Since $B$ is path-connected, choose a path $\gamma$ from $b$ to $b^{\prime }$. Horizontality of $\mu$ and part c implyThus $\mu (b^{\prime })$ is conjugate to the isomorphism $\mu (b)$ and is itself an isomorphism. Since $b^{\prime }$ was arbitrary, $\mu$ is fiberwise invertible everywhere.