Let $F:M\to N$ be a map of smooth manifolds. Show that the following are equivalent:
- $F$ is smooth,
- For each $p\in M$ there exist smooth charts $(U,\varphi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $U\cap F^{-1}(V)$ is open in $M$ and the composite map $\psi\circ F\circ \varphi^{-1}$ is smooth from $\varphi(U\cap F^{-1}(V))$ to $\psi(V)$,
- $F$ is continuous and there exist smooth atlases $\{(U_\alpha,\varphi_\alpha)\}$ and $\{(V_\beta,\psi_\beta)\}$ for $M$ and $N$, respectively, such that for each $\alpha$ and $\beta$, $\psi_\beta\circ F\circ \varphi_\alpha^{-1}$ is smooth from $\varphi_\alpha(U_\alpha\cap F^{-1}(V_\beta))$ to $\psi_\beta(V_\beta)$.
For those interested, this is exercise 2.7 from Lee's text on Smooth Manifolds (pg35).
I can show 1 $\iff$ 2, and 3 $\implies 2$. For 2 $\implies$ 3, I can show that $F$ is continuous. By hypothesis we get a smooth atlas $\{(U_p,\varphi_p)\}_{p\in M}$ for $M$ and a collection of smooth charts $\mathcal N:=\{(V_p,\psi_p)\}_{p\in M}$ for $N$ satisfying the desired conclusion. But $\mathcal N$ isn't necessarily a smooth atlas for $N$ since the domains don't necessarily cover $N$.
I feel like I should be able to take the given smooth structure for $N$ since $\mathcal N$ will (uniquely) determine this structure anyways. But, how do I know that the coordinate representations will satisfy the desired conclusion (assuming this approach works, of course)?
Or is there some way to "extend" $\mathcal N$ to an atlas (not necessarily maximal) for $N$ with the desired property?