  Given an exact Lagrangian submanifold $L$ in $T^*N$, we want to construct a
complex of sheaves in the derived category of sheaves on $N\times {\mathbb R}
$, such that its singular support, $SS({\mathcal F}^\bullet_L)$, is equal to
$\widehat L$, the cone constructed over $L$. Its existence was stated in
\cite{Viterbo-ISTST} in 2011, with a sketch of proof, which however contained a
gap (fixed here by the rectification). A complete proof was shortly after
provided by Guillermou (\cite{Guillermou}) by a completely different method, in
particular Guillermou's method does not use Floer theory. The proof provided
here is, as originally planned, based on Floer homology. Besides the
construction of the complex of sheaves, we prove that the filtered versions of
sheaf cohomology of the quantization and of Floer cohomology coincide, that is
$FH^*(N\times ]-\infty, \lambda [, {\mathcal F}^\bullet_L)\simeq
FH^*(L;0_N;\lambda)$, and so do their product structures.
