Let $h:\omega\to \omega$ is a monotone increasing function that goes to infinity.
For $n<\omega$ let $\mathcal{S}_{n}(\omega,h)=\{s:n \to{\omega}^{<\omega}: \forall i< \omega(|s(i)|\leq h(i))\}$ and let $\mathcal{S}_{<\omega}(\omega,h)=\bigcup_{n<\omega}\mathcal{S}_{n}(\omega,h)$
The localization forcing for $h$ is $$\mathbb{LOC}_h=\{(s,F):s \in \mathcal{S}_{<\omega}(\omega,h),F \subseteq \omega^{\omega} \text{ and } |F|\leq h(|s|)\},$$ with the order defined by $(s,F)\leq (s',F')$ iff $s \subseteq s'$, $F \subseteq F'$ and $\forall_{i\in|s'|\setminus|s|}\forall_{x \in F}(x(i) \in s'(i))$.
I know that $\mathbb{LOC}_h$ generically adds a new $S_G=\bigcup\{s:\exists F((s,F) \in G\}$, where $G$ is the generic filter over $V$.
How can I show that $V[G]=V[S_G]$?