Let $M,N$ be smooth manifolds and a function $f:M\to N$.
By definition, $f$ is smooth when $\psi\circ f\circ\varphi^{-1}$ is smooth for every chart $\varphi$ of $M$ and $\psi$ of $N$.
But I've noticed this: for different charts $\widetilde{\varphi}$ of $M$ and $\widetilde{\psi}$ of $N$, we have that $\widetilde{\psi}\circ\psi^{-1}$, $\varphi\circ\widetilde{\varphi}^{-1}$ are smooth, and that:
$$\widetilde{\psi}\circ f\circ\widetilde{\varphi}^{-1}=(\widetilde{\psi}\circ\psi^{-1})\circ(\psi\circ f\circ\varphi^{-1})\circ(\varphi\circ\widetilde{\varphi}^{-1})$$
So if we know $\psi\circ f\circ\varphi^{-1}$ is smooth, it follows that $\widetilde{\psi}\circ f\circ\widetilde{\varphi}^{-1}$ is smooth. So it seems that I've shown that:
In order to check whether $f$ is smooth, it's enough to take only one chart $\varphi$ of $M$ and only one chart $\psi$ of $N$ and verify that $\psi\circ f\circ\varphi^{-1}$ is smooth.
This looks very strange, because if this is true, I only need to know $M,N$ locally in order to conclude something about $f$, which generally involves $M,N$ globally.
Am I missing something?