  Given a smooth projective toric variety $X_\Sigma$ of complex dimension $n$,
Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of
the differential graded (dg) derived category of coherent sheaves
$Coh(X_\Sigma)$ into the dg derived category of constructible sheaves on a
torus $Sh(T^n, \Lambda_\Sigma)$. Recently, Kuwagaki \cite{Ku2} proved that the
quasi-embedding is a quasi-equivalence, and generalized the result to toric
stacks. Here we give a different proof in the smooth projective case, using
non-characteristic deformation of sheaves to find twisted polytope sheaves that
co-represent the stalk functors.
