Notes on Smooth Manifolds and Vector Bundles
... is a diffeomorphism for every (U, ϕ) ∈ FM . A composition of two smooth maps (local diffeomorphisms, diffeomorphisms) is again smooth (a local diffeomorphism, a diffeomorphism).
It is generally impractical to verify that the map (2.1) is smooth for all (U, ϕ) ∈ FM and (V, ψ) ∈ FN .
The following lem ...