Definition. A lattice in $\mathbb{C}$ is a subgroup $L$ of $\mathbb{C}$ of the following form: for some $\omega_{1},\omega_{2}\in\mathbb{C}\setminus\{0\}$ with $\omega_{1}/\omega_{2}\not\in\mathbb{R},$ $$L=\{n_{1}\omega_{1}+n_{2}\omega_{2}:n_{1},n_{2}\in\mathbb{Z}\}.$$ If $\gamma\in\mathbb{C}$ and $L$ is a lattice as above, then we define $$\gamma L = \{\gamma\omega:\omega\in L\}.$$
Question. Let $L$ and $M$ be lattices in $\mathbb{C}.$ Suppose $\gamma,\delta\in\mathbb{C}$ are non-zero and such that $\gamma L\subseteq M$ and $\delta M\subseteq L.$ Does it then follow that $\gamma L=M?$
Context. I am working through Rick Miranda's Algebraic Curves and Riemann Surfaces. In Chapter III he considers holomorphic maps between complex tori, and appears to implicitly assume that the queried implication holds, but I can't see why it should be true.
My thoughts so far. I could try contradiction. For example, if $\gamma L\subset M,$ then $\gamma\delta L\subset L,$ and I think it follows that $\lvert\gamma\delta\rvert>1,$ but I don't see where to go from here.
How about $L=\Bbb Z+\Bbb Z i$, $M=\Bbb Z+\Bbb Z(2i)$, $\gamma=2$ and $\delta=1$?