I was reading this proposition in Lee's book and I had two questions concerning it.
Proposition 5.2 (Images of Embeddings as Submanifolds). Suppose $M$ is a smooth manifold with or without boundary, $N$ is a smooth manifold, and $F : N \to M$ is a smooth embedding. Let $S = F(N)$ With the subspace topology, $S$ is a topological manifold, and it has a unique smooth structure making it into an embedded submanifold of $M$ with the property that $F$ is a diffeomorphism onto its image.
Proof. If we give S the subspace topology that it inherits from $M$; then the assumption that $F$ is an embedding means that $F$ can be considered as a homeomorphism from $N$ onto $S$ , and thus $S$ is a topological manifold. We give S a smooth structure by taking the smooth charts to be those of the form $(F(U),\phi \circ F^{-1})$, where $(U,\phi)$ is any smooth chart for N ; smooth compatibility of these charts follows immedi- ately from the smooth compatibility of the corresponding charts for $N$. With this smooth structure on $S$ , the map $F$ is a diffeomorphism onto its image (essentially by definition), and this is obviously the only smooth structure with this property. The inclusion map $S \hookrightarrow M$ M is equal to the composition of a diffeomorphism followed by a smooth embedding: $S \xrightarrow{F^{-1}} N \xrightarrow{F} M$ - and therefore it is a smooth embedding.
How do we show that
this is obviously the only smooth structure with this property
Also if we take the basic example where $F=\iota$ the inclusion map, then the structure we define on $S$ should be $(\iota^{-1}(U),\phi \circ \iota)= (U\cap S, \phi|_S)$ no? And not $(\iota(U),\phi \circ \iota^{-1})$ right?
Assume that $F(N)$ has a smooth structure such that $F$ is diffeomorphism onto its image. If $(U,\phi)$ be a smooth chart on $N$, then our assumption implies that $U$ is diffeomorphic with $F(U)$. Since $(U,\phi)$ is a smooth chart, $U$ and $\phi(U)$ are diffeomorphic. Therefore $F(U)$ and $\phi(U)$ are diffeomorphic via $\phi\circ F^{-1}$. This defines a smooth chart of $F(N)$. If we have an open cover of $N$ by smooth charts, say $\{(U_i,\phi_i)\}_{i\in I}$, then $\{(F(U_i),\phi_i\circ F^{-1})\}_{i\in I}$ is an open cover of $F(N)$. Obviously these charts are compatible, so they form a smooth atlas for $F(N)$.
Indeed, what I've proved is: assume that $F(N)$ has already given a smooth structure with the desired property and $(U,\phi)$ a smooth chart of $N$, then $(F(U),\phi\circ F^{-1})$ is a smooth chart in this smooth structure. Smooth charts of this form covers $F(N)$, so they define the given smooth structure as well. Hence the smooth structure we've assumed must be given by charts of the form $(F(U),\phi\circ F^{-1})$.
In your example, it should be less confused if you specify your domain and target. You want to have a smooth chart of $\iota(N)$, not $N$. For $U$ is a chart in $N$ then the corresponded chart in $\iota(N)$ is $\iota(U)$. You're taking $U$ as a chart in $M$ and deduce a chart $\iota^{-1}(U)$ in $N$.