In the exercise 2.13.4 of the book Meccanica Analitica: Meccanica Classica, Meccanica Lagrangiana e Hamiltoniana e Teoria della Stabilità by Valter Moretti, a concept of diffeomorphism is defined for a map $\psi: A^n \to A^m$ between two affine spaces: being bijective and both $\psi,\psi^{-1}$ represented by Cartesian coordinate systems (also called affine coordinate systems in other books) as functions from $\mathbf{R}^n$ to $\mathbf{R}^m$ and $\mathbf{R}^m$ to $\mathbf{R}^n$, respectively being smooth in the standard sense.
However, affine spaces can also be treated as particular types of smooth manifolds and so are equipped with smooth structures, which we use the define differentiability of a map by a method quite similar to the "affine method", but also using local coordinate systems that are in the atlas.
I can't see the equivalence between defining differentiability in the standard "manifold sense" and defining by requiring only differentiability with respect to Cartesian coordinate systems. Can someone help me?