Let $\Gamma,\Gamma^\prime\subset\mathbb{C}$ be two lattices, and let $f:\mathbb{C}/\Gamma\to\mathbb{C}/\Gamma^\prime$ be a non-constant holomorphic map such that $f(0)=0$. One can show that that there exists a unique map $F:\mathbb{C}\to\mathbb{C}$ given by $F(z)=\alpha z$ for some $\alpha\in\mathbb{C}^\ast$, such that $\pi^\prime\circ F=f\circ\pi$, where $\pi:\mathbb{C}\to\mathbb{C}/\Gamma$ and $\pi^\prime:\mathbb{C}\to\mathbb{C}/\Gamma^\prime$ are the canonical projections, and that then $\alpha\Gamma\subset\Gamma^\prime$ holds.
I have also shown that $f$ is an unbranched covering map, and that each $\gamma^\prime\in\Gamma^\prime$ defines an element $p_{\gamma^\prime}$ of $Deck(\mathbb{C}/\Gamma\to\mathbb{C}/\Gamma^\prime)$ by the map sending $\pi(z)\in\mathbb{C}/\Gamma$ to $\pi(z+\gamma^\prime\alpha^{-1})$. Moreover since each for each $\alpha\gamma\in\alpha\Gamma\subset\Gamma^\prime$ it holds that $p_{\alpha\gamma}=id_{\mathbb{C}/\Gamma}$, it follows that we have a well-defined map from $\Gamma/\alpha\Gamma^\prime$ to $Deck(\mathbb{C}/\Gamma\to\mathbb{C}/\Gamma^\prime)$ which makes $\Gamma^\prime/\alpha\Gamma$ into a subgroup of $Deck(\mathbb{C}/\Gamma\to\mathbb{C}/\Gamma^\prime)$.
I am trying to show that this map is an isomorphism, i.e. that each deck transformation is of the form $p_{\gamma^\prime}$, but I am stuck. I know that if $F$ is a deck transformation then $f(F(z))=f(z)$ for all $z\in\mathbb{C}$, hence taking $z=0$ and noting that $f(0)=0$ implies that $f(F(0))\in\Gamma^\prime$, but I don't know where to go from here. Any help will be appreciated (exercise is from Forster & Gilligan)