Let $M$ be a smooth manifold with $\mathscr A=\{(U_\alpha,\varphi_\alpha)\}_{\alpha\in J}$ being its smooth structure.
Let $V$ be open in $M$ and $(U,\varphi)\in \mathscr A$.
Define $\psi:U\cap V\to \varphi(U\cap V)$ as $\psi(p)=\varphi(p)$ for all $p\in V$.
We note that $(V\cap U,\psi)$ is a chart on $M$.
QUESTION: Does $(V\cap U,\psi)\in\mathscr A$?
I think the answer to the above question is yes because $(V\cap U,\psi)$ is smoothly compatible with every member of $\mathscr A$. Thus $(V\cap U,\psi)\in\mathscr A$.
Can somebody confirm that I am correct or else point out a mistake?
PS: I am an absolute beginner in Differential Geometry.
Thank you.