Let $V$ be a finite vector space over $\mathbb{C}$ and consider a lattice $L$ of $V$ i.e a discrete subgroup of $V$ of maximal rank. Consider the torus $T=V/L$.
Definition: Let $S\subset T$ be a subset of $T$. We say that $S$ is a subtorus of $T$ if there exist a subspace $W\le V$ and a lattice $M$ on $W$ such that $M\subset W \cap L$.
My question: I think that this definition does not make sense because any subset of the quotient $T=V/L$ is a set of the form $A/L$ for $A\subset V$. But changing the quotient "is not allowed". I know that there are natural identifications in some cases but they are never the equality of sets.
I need to know this because I read a proof of the following proposition:
Proposition: Let $T_i= V_i/L_i$ be complex torus and let $f:T_1\to T_2$ be a homomorphism betweem them, then $Im f$ is a subtorus of $T_2$.
The proof uses the lift of the function $f$ i.e the analityc representation $F:V_1 \to V_2$. Then the book says that the following equality holds $Im f = F(V_1) / \left(F(V_1) \cap L_2 \right)$.
Here again I don't know how the set $F(V_1) / \left(F(V_1) \cap L_2 \right)$ can be considered as a subset of $V_2/L_2$ all my problems come from the quotient