I'm confused about the theorem cited below from Guillemin-Pollack's "Differential Topology".
Why do they refer to the maps $\phi: W\rightarrow X$ and $\psi: U\rightarrow Y$ as local parametrizations? By definition, local parametrizations are diffeomorphisms. But $\phi, \psi$ need not be onto. (Since $X$ is a manifold, there is a neighborhood $O_x\subset X$ of a given point $x$ such that $W\rightarrow O_x\subset X$ is a diffeomorphism (with $W$ as in the text); and this map is a parametrization.)
In particular, it is unclear how we can even define $(\phi\times \psi)^{-1} $ since $\phi\times \psi$ need not be onto.
