smoothness of an inclusion map proof verification

Question

I know this problem was asked before. My solution is different. I want to verify my solution.

Let $M$ and $N$ be manifolds and let $q_0$ be a point in $N$. Prove that the inclusion map $i_{q_0} : M \to M×N : p \mapsto (p,q_0)$, is $C^\infty$.

Solution attempt:

Let $(U,\phi)$ and $(V,\psi^{\prime})$ be charts on M and $M \times N$ respectively. Charts on $M \times N$ are of the form $(U_1\times V_1, \psi_1 \times \psi_2)$. Therefore we can take the following chart $(U \times V_1, \phi \times \psi_2) = (U \times V_1,\psi^{\prime})$.

It follows that:

$(\psi^{\prime} \circ F \circ \phi^{-1})(\phi(p)) = (\psi^{\prime} \circ F)(p) = \psi^{\prime}((p,q_0)) = (\phi(p),\psi_2(q_0))$.

Since components $\psi_1$ and $\psi_2$ are $C^{\infty}$ it follows F is $C^{\infty}$.

Answer 1

I know this problem was asked before. My solution is different. I want to verify my solution.

Let $M$ and $N$ be manifolds and let $q_0$ be a point in $N$. Prove that the inclusion map $i_{q_0} : M \to M×N : p \mapsto (p,q_0)$, is $C^\infty$.

Solution attempt:

Let $(U,\phi)$ and $(V,\psi^{\prime})$ be charts on M and $M \times N$ respectively. Charts on $M \times N$ are of the form $(U_1\times V_1, \psi_1 \times \psi_2)$. Therefore we can take the following chart $(U \times V_1, \phi \times \psi_2) = (U \times V_1,\psi^{\prime})$.

It follows that:

$(\psi^{\prime} \circ F \circ \phi^{-1})(\phi(p)) = (\psi^{\prime} \circ F)(p) = \psi^{\prime}((p,q_0)) = (\phi(p),\psi_2(q_0))$.

Since components $\psi_1$ and $\psi_2$ are $C^{\infty}$ it follows F is $C^{\infty}$.