I'm trying to solve the following question:
Let $F:M\longrightarrow N$ be a bijective map. Prove that, if M is an $n$-dimensional differentiable manifold, then $N$ admits a unique differentiable structure making $F$ a diffeomorphism.
I think that the differentiable structure we are looking for would be given by the atlas $A=\{(F(U_a), F|_{U_a})\}_{a\in T}$, where $T$ is the topology induced in N by F, forcing $F$ to be a homeomorphism. But how can I prove the uniqueness?
I'm identifying a differentiable structure with a maximal atlas. Suppose you have another maximal atlas $A'$ on $M$, such that $F$ is a diffeomorphism between $(M,A')$ and $N$. To say that $A'$ and $A$ give the same differentiable structure is to say that they are compatible atlases. So for a chart $U',\phi'$ in $A'$ and $U,\phi$ in $A$, you need to show that the bijection
$$\phi'(U' \cap U) \xleftarrow{\phi'} U' \cap U \xleftarrow{\phi^{-1}} \phi U' \cap U$$
is a diffeomorphism. Since this is a local question, we are dealing with maximal atlases, and every open subset of a chart inherits the structure of a chart, we may shrink our open sets and assume $U' = U$, and moreover that $V = F(U) = F(U')$ together with some map $\psi$ is a chart on $V$.
Since $F$ is a diffeomorphism with respect to the maximal atlases $A$ and $A'$, we know that $\psi \circ F \circ (\phi')^{-1}$ and $\psi \circ F^{-1} \circ \phi^{-1}$ are diffeomorphisms. So is $$ (\psi \circ F \circ (\phi')^{-1})^{-1} \circ (\psi \circ F \circ \phi^{-1}) = \phi' \circ \phi^{-1}$$
