= Solution
Outside the finite set of <branch points>, the derivative is nonzero. The <inverse function theorem> supplies evenly covered neighbourhoods, and compactness of the <Riemann sphere> makes every fibre finite, so the restriction is a covering map.
Lift a loop $γ$ from each point of the fibre over $z_0$; its endpoints define a <permutation> $σ_γ$. The homotopy lifting property shows that this permutation depends only on the based <homotopy> class. The punctured source is connected, so a path between any two points in the fibre projects to a loop whose lift joins them. Hence the monodromy group acts transitively.
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