A map $\phi: M\to N$ is smooth if for sufficiently many coordinates systems $\eta$ cover $N$ the functions $y_i\circ\phi$ are smooth.
I tried to prove this but I would like someone to see if this is correctly written as a proof.
Taking $p=(p_1,...,p_n)$ in $M$, then $y_i\circ\phi\circ x_i^{-1}$ is smooth for the coordinate $i$. The same happens for the other coordinates. So $\phi$ is smooth.
Question: Is this correct? How do I relate this to the coordinate systems $\eta$ in $N$?