= Solution
Choose an actual pole $p_i$ of $f$; one exists by nonconstancy and compactness. In a local coordinate $z$ centered there, its <simple pole> has expansion $f(z)=a_{-1}/z+O(1)$ with $a_{-1}\ne0$. Hence
$$
w(z)=1/f(z)=z/a_{-1}+O(z^2),\qquad w'(0)\ne0.
$$
The <holomorphic inverse function theorem> makes \b[$1/f$ a local coordinate]. Here the source notation $f^{-1}$ denotes the reciprocal, rather than an inverse map of the <meromorphic function>.
The chosen pole is among the fixed points, so $\varphi$ maps a sufficiently small neighborhood of it into the coordinate neighborhood. If $f=f\circ\varphi$, then $w=w\circ\varphi$ there. Injectivity of this coordinate forces $\varphi(x)=x$ throughout a nonempty open set. The stated identity principle then gives \b[$\varphi$ equal to the identity everywhere].
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