In the paper "Symplectic manifolds and their Lagrangian submanifolds", Weinstein showed that locally all the Lagrangian foliations are symplectomorhic to the fiber foliation of a cotangent bundle.

I have a tiny problem in one step in his argument. Let $N$ be a smooth manifold and $U$ a small neighborhood of the zero section of $T^*N$ with the canonical symplectic structure on $T^*N$. Let $F$ be a Lagrangian foliation on $U$ such that it is transversal to $N$ at each point of $N$. Weinstein asserts that there exist a diffeomorphism $h$ from a neighborhood of $N$ to some other neighborhood of $N$ such that $h$ is the identity map on $N$ and $h$ maps leaves of $F$ onto cotangent fibers. I am wondering how this $h$ could be constructed. Probably it has something to do with the Whitney extension theorem. Maybe this is something really standard, I would like to know the details anyway.

Thank you for your attention.