Let $\Gamma$ and $\Gamma'$ be two lattices in $\mathbb{C}$ such that $\Gamma \subseteq \Gamma'$. Show that the natural map $\mathbb{C}/\Gamma \to \mathbb{C}/\Gamma'$ is holomorphic.
The map $f: \mathbb{C}/\Gamma \to \mathbb{C}/\Gamma'$ is holomorphic if and only if $f \circ \pi_1 : \mathbb{C} \to \mathbb{C}/\Gamma'$ is holomorphic where $\pi_1: \mathbb{C} \to \mathbb{C}/L$.
Now we know that the projections are holomorphic so what I want to deduce is that $ f\circ \pi_1 = \pi_2$ where $\pi_2$ is the projection $\mathbb{C} \to \mathbb{C}/\Gamma'$. Is this true?