In Nakahara's GTaP chapter 8.1, he introduces the lattice $L(\omega_1,\omega_2)$ generated by two complex numbers $\omega_i$ (where $\text{Im}(\omega_1/\omega_2)>0$). Then he defines a complex structure on $T^2$ by introducing the equivalence relation of complex numbers $z_1\sim z_2$ iff $z_1=z_2+n\omega_1+m\omega_2$ with $m,n\in\mathbb Z$, so $\mathbb C/L(\omega_1,\omega_2)$ is $T^2$ as a complex manifold.
Then he examines the complex structures of two different pairs of complex numbers $(\omega_1,\omega_2)$ and $(\tilde\omega_1,\tilde\omega_2)$:
Assume that there exists a one-to-one holomorphic map $h$ of $\mathbb C/L(\omega_1,\omega_2)$ onto $\mathbb C/L(\tilde\omega_1,\tilde\omega_2)$ where $\text{Im}(\omega_1/\omega_2)>0$, $\text{Im}(\tilde\omega_1/\tilde\omega_2)>0$. Let $p:\mathbb C\to\mathbb C/L(\omega_1,\omega_2)$ and $\tilde p:\mathbb C\to\mathbb C/L(\tilde\omega_1,\tilde\omega_2)$ be the natural projections. [...] Choose the origin $0$ and define $h_*(0)$ to be a point such that $\tilde p\circ h_*(0)=h\circ p(0)$.
Then by analytic continuation from the origin, we obtain a one-to-one holomorphic map $h_*$ of $\mathbb C$ onto itself satifying $\tilde p\circ h_*(0)=h\circ p(0)$.
Why do we know that $h_*$ is one-to-one $\mathbb C\to\mathbb C$? Why aren't there singularities? Why is $h_*$ surjective onto $\mathbb C$?