In physics, we often use diffeomorphisms to change coordinates on a smooth manifold $(M,A)$. But, from what I've seen, "changing coordinate" simply corresponds to give our self another atlas $B$ wich is compatible with $A$. Then, if \begin{align*} A &= \{(U_i,\phi_i)\}_{i\in I},\\ B &= \{(V_i,\psi_i)\}_{i\in J},\\ \end{align*} we locally have the following smooth change of coordinates: \begin{equation} \psi_j\circ\phi^{-1}_i:\phi_i(U_i\cap V_j)\to\psi_j(U_i\cap V_j). \end{equation} Thus, the maps \begin{equation} \psi^{-1}_j\circ\phi_i:U_i\cap V_j\to U_i\cap V_j \end{equation} are diffeomorphisms between $U_i\cap V_j$ and itself. Is that right?
But is it possible to create a global diffeomorphism from those?
This question is related to the fact that I don't really understand why GR is a gauge theory.