The claim was made in my textbook as a part of a proof of Minkowski's theorem without any explanation. Intuively it feels correct, but that's not enough in math.
The map in question is $$\frac{1}{2} S\rightarrow\mathcal{F},\ \frac{1}{2} a\rightarrow w_{\frac{1}{2} a}.$$ Here $S\subset \mathbb{R}^n$ is a bounded symmetric convex set and $w\in\mathcal{F}$ is in the fundamental parallelpiped of lattice $L$, so it is also a bounded, symmetric and convex subset of $\mathbb{R}^n$. Vector $a$ is of course in $S$. Every vector $t \in \mathbb{R}^n$ can be written as $t=v_t+w_t$ where $v_t\in L,\ w_t\in\mathcal{F}$, so that's what the subscript is in reference to.
I don't know if the map matters though because the wording makes it seem that the claim should be generally true.