Let $U$ and $V$ be open subsets of a Hausdorff topological space $M$. Let $\phi$ be a homeomorphism from $U$ to an open subset of $\mathbb{R}^n$, and $\psi$ be a homeomorphism from $V$ to an open subset of $\mathbb{R}^m$, and $U \cap V \neq \emptyset$.
What does the following notation mean:
$\psi \circ \phi^{-1} = \phi(U \cap V) \subset \mathbb{R}^n \rightarrow \psi(U \subset V) \subset \mathbb{R}^m$
I would expect that $\psi \circ \phi^{-1}: V \rightarrow \mathbb{R}^m \rightarrow \mathbb{R}^n \rightarrow U$. So what does the above notation mean?
EDIT: I made a mistake if $f: X \rightarrow Y$, and $g: Y \rightarrow Z$...then $g(f(x)) = (g \circ f )(x)$
Just to elaborate on what's already been said, here $U,V \subset M$. If you consider $\psi \circ \phi^{-1}$ then you would think $\psi \circ \phi^{-1}: \phi(U) \to \psi(\phi(U))$ which is somewhat correct (* I'll say more soon). If $\phi^{-1}(p) \in U \setminus V$ (which it can be) then $\psi(\phi^{-1}(p))$ is not defined. Hence, we must take $p \in \phi(U \cap V)$ for the composition to make sense. Therefore we have;
$$\psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V) $$
I think it is abuse of notation to have $U \subset V$ in the space you are mapping to. I think what is meant here is that you can reduce to a subset which is contained in $V$. Lastly, the remark about * refers to the fact that $f(x) = x^2$ is a map from $\mathbb{R} \to \mathbb{R}^+ \subset \mathbb{R}$ so it is a map from $\mathbb{R}$ into $\mathbb{R}$.