Suppose that $f: \mathbb{C} / \Lambda_{1} \rightarrow \mathbb{C} / \Lambda_{2}$ is a holomorphic map of complex tori, and let $\pi_{j}$ denote the projection map $\mathbb{C} \rightarrow \mathbb{C} / \Lambda_{j}$ for $j=1,2$. Show that there is a holomorphic map $F: \mathbb{C} \rightarrow \mathbb{C}$ such that $\pi_{2} F=f \pi_{1} .$

Prove that $F(z)=\lambda z+\mu$ for some $\lambda, \mu \in \mathbb{C}$. Hence deduce that two complex tori $\mathbb{C} / \Lambda_{1}$ and $\mathbb{C} / \Lambda_{2}$ are conformally equivalent if and only if the lattices are related by $\Lambda_{2}=\lambda \Lambda_{1}$ for some $\lambda \in \mathbb{C}^{*}$.