Let $L_1,L_2$ be two lattices of rank 2 in $\mathbb C$. Then $\mathbb C/L_1$ and $\mathbb C/L_2$ are two tori. Let $$f:\mathbb C/L_1\to\mathbb C/L_2$$ be a conformal equivalence. Show that the lift $\tilde f:\mathbb C\to\mathbb C$ of $f$ is an automorphism (conformal equivalence to itself) of $\mathbb C$.
My attempt: Donote the natural projections by $$\pi_1:\mathbb C\to\mathbb C/L_1$$ $$\pi_2:\mathbb C\to\mathbb C/L_2$$ Then the lift $\tilde f$ satisfies $$f\circ\pi_1=\pi_2\circ\tilde f$$ and is holomorphic. The same applies to $f^{-1}$. $$f^{-1}\circ\pi_2=\pi_1\circ\widetilde{f^{-1}}$$ $$\implies \pi_2=(ff^{-1})\pi_2=f\pi_1\widetilde{f^{-1}}=\pi_2\tilde f\widetilde{f^{-1}}$$ But I can't cancel $\pi_2$ as it is not injective.
What am I missing?
You use the uniqueness of lifts: $\tilde{f}\widetilde{f^{-1}}$ and $Id_{\mathbb{C}}$ are both lifts of the map $\pi_2$ along $\pi_2$ which preserve the basepoint, so they must be equal.