I want to prove that if $F: M\rightarrow N$ is smooth, where $M,N$ are smooth manifolds, then its coordinate representation w.r.t any smooth charts $(U,\phi),~ (V,\psi)$ such that $F(U)\subseteq V$ is smooth.
By definition we know that for all $p\in M$ there are smooth charts $(U,\phi), (V,\psi)$ of $M,N$ respectively with $p\in U$ and $F(p)\in V$ such that $F(U)\subseteq V$ and the coordinate representation $\psi\circ F\circ \phi^{-1}: \phi(U) \rightarrow \psi(V) $ is smooth.
My idea was to take arbitrary smooth charts $(U,\phi),~ (V,\psi)$ of $M,N$ respectively such that $F(U)\subseteq V$. Now by definition for all $p \in U$, we get smooth charts $(\tilde{U}_p,\tilde{\phi}_p), (\tilde{V}_p,\tilde{\psi}_p)$ of $M,N$ respectively with $p\in \tilde{U}_p$ and $F(p)\in \tilde{V}_p$ such that $F(\tilde{U}_p)\subseteq \tilde{V}_p$ and $\tilde{\psi}_p \circ F \circ \tilde{\phi}_p^{-1}$ is smooth.
We know that $U\cap \tilde{U}_p\neq\emptyset$ and $V\cap \tilde{V}_p\neq\emptyset$. Hence, have that $(U\cap \tilde{U}_p, \tilde{\phi}_p \vert_{U\cap \tilde{U}_p})$ and $(V\cap \tilde{V}_p, \tilde{\psi}_p \vert_{V\cap \tilde{V}_p})$ are smooth charts and since the the transition functions $\phi \circ \tilde{\phi}_p^{-1}$ and $\psi \circ \tilde{\psi}_p^{-1}$ are diffeomorphisms, from the smoothness of $\tilde{\psi}_p \circ F \circ \tilde{\phi}_p^{-1}$ follows that the $\psi_p\vert_{V\cap \tilde{V}_p} \circ F \circ \phi_p\vert_{U\cap {U}_p}^{-1}$ is smooth. Since, this holds for all $p\in U$ we can take the union of all charts to obtain the claim.
I'm not sure if my proof is really valid because especially the last part where I take the union of the constructed charts. Also I somehow think that there should be a simpler argument for this proof.
According to your definition, if $F$ is smooth around $p$ then there exist charts $(U,\phi)$ around $p$ and $(V,\psi)$ around $F(p)$ such that the map, \begin{equation} \psi\circ F\circ\phi^{-1}: \phi(U)\to \psi(U) \end{equation} is smooth.
We'll now show it is smooth for any choice of charts around $p$ and $F(p)$. Take, $(\tilde{U}, \tilde{\phi})$ chart around $p$ and $(\tilde{V}, \tilde{\psi})$ chart around $F(p)$ then, on $\tilde{\phi}(U\cap\tilde{U})$ we have; \begin{equation} \tilde{\psi}\circ F\circ \tilde{\phi}^{-1}= (\tilde{\psi}\circ\psi^{-1})\circ(\psi\circ F\circ \phi^{-1})\circ(\phi\circ\tilde{\phi}^{-1}). \end{equation} We know that the map in the middle is smooth. Also the other two maps are smooth by definition of an atlas, so the whole thing is smooth which demonstrates that it doesn't matter which chart we use.