[proofplan]
We restrict the two chart [homeomorphisms](/page/Homeomorphism) to the overlap $U\cap V$, allowing the overlap to be empty. The restriction of $\varphi$ identifies $U\cap V$ homeomorphically with $\varphi(U\cap V)$, and the restriction of $\psi$ identifies the same overlap homeomorphically with $\psi(U\cap V)$. The transition map is exactly the composition of the inverse of the first restricted map with the second restricted map, so it is a homeomorphism by a direct verification of bijectivity, continuity, and continuity of the inverse.
[/proofplan]
[step:Restrict each chart to the overlap]
Let $W:=U\cap V$, regarded as a subspace of $M$. Since $U$ and $V$ are [open](/page/Open%20Set) in $M$, the set $W$ is open in both $U$ and $V$ with their [subspace topologies](/page/Subspace%20Topology).
Define the restricted chart maps $\varphi_W:W\to\varphi(W)$ by $\varphi_W(p):=\varphi(p)$ for every $p\in W$, and $\psi_W:W\to\psi(W)$ by $\psi_W(p):=\psi(p)$ for every $p\in W$. Because $\varphi:U\to\varphi(U)$ and $\psi:V\to\psi(V)$ are bijections onto their images, the restrictions $\varphi_W$ and $\psi_W$ are bijections onto $\varphi(W)$ and $\psi(W)$, respectively.
We claim that $\varphi_W$ is a homeomorphism. Its continuity follows from the continuity of $\varphi$ by restriction of domain and codomain. Its inverse is the restriction
\begin{align*}
\varphi^{-1}|_{\varphi(W)}:\varphi(W)\to W
\end{align*}
of the continuous inverse $\varphi^{-1}:\varphi(U)\to U$, so it is continuous. Hence $\varphi_W$ is a homeomorphism. For $\psi_W$, continuity follows from the continuity of $\psi:V\to\psi(V)$ by restriction of domain and codomain. Its inverse is the restriction $\psi^{-1}|_{\psi(W)}:\psi(W)\to W$ of the continuous inverse $\psi^{-1}:\psi(V)\to V$, so it is continuous. Hence $\psi_W$ is a homeomorphism.
[guided]
The point of this step is to replace the original chart domains $U$ and $V$ by the common domain on which both coordinate maps are defined. Let
$W:=U\cap V$, with the topology inherited from $M$. Since $U$ and $V$ are open subsets of $M$, the intersection $W$ is open in $M$, and therefore $W$ is also open as a subspace of both $U$ and $V$.
We define two maps on this common domain: $\varphi_W:W\to\varphi(W)$ by $\varphi_W(p):=\varphi(p)$ for every $p\in W$, and $\psi_W:W\to\psi(W)$ by $\psi_W(p):=\psi(p)$ for every $p\in W$.
These are not new coordinate maps; they are the old charts restricted to the overlap. Because $\varphi:U\to\varphi(U)$ is injective, its restriction $\varphi_W$ is injective. Because the codomain of $\varphi_W$ is exactly $\varphi(W)$, the map $\varphi_W$ is surjective onto its stated codomain. Thus $\varphi_W$ is bijective. The same argument shows that $\psi_W$ is bijective.
We now verify the topological part. Since $\varphi:U\to\varphi(U)$ is continuous, the restriction $\varphi_W:W\to\varphi(W)$ is continuous when $W$ and $\varphi(W)$ carry their subspace topologies. Its inverse is precisely
\begin{align*}
\varphi_W^{-1}:\varphi(W)\to W
\end{align*}
where $\varphi_W^{-1}(x)=\varphi^{-1}(x)$ for every $x\in\varphi(W)$. This is the restriction of the continuous map $\varphi^{-1}:\varphi(U)\to U$ to the subspace $\varphi(W)\subset\varphi(U)$, with image lying in $W\subset U$, so it is continuous. Therefore $\varphi_W$ is a homeomorphism. For $\psi_W$, the map is continuous because it is the restriction of the continuous map $\psi:V\to\psi(V)$. Its inverse is $\psi_W^{-1}:\psi(W)\to W$, where $\psi_W^{-1}(y)=\psi^{-1}(y)$ for every $y\in\psi(W)$. This is the restriction of the continuous map $\psi^{-1}:\psi(V)\to V$ to the subspace $\psi(W)\subset\psi(V)$, with image lying in $W\subset V$, so it is continuous. Hence $\psi_W:W\to\psi(W)$ is also a homeomorphism.
[/guided]
[/step]
[step:Identify the transition map as a composition of the restricted homeomorphisms]
Consider the map $T:\varphi(W)\to\psi(W)$ defined by $T:=\psi_W\circ\varphi_W^{-1}$. For every $x\in\varphi(W)$, the point $\varphi_W^{-1}(x)$ is the unique point $p\in W$ such that $\varphi(p)=x$. Hence
\begin{align*}
T(x)=\psi(\varphi^{-1}(x)).
\end{align*}
Thus $T$ is exactly the transition map $\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)$.
[/step]
[step:Prove that the transition map is a homeomorphism]
Since $\varphi_W$ and $\psi_W$ are homeomorphisms, the inverse $\varphi_W^{-1}:\varphi(W)\to W$ is continuous, and $\psi_W:W\to\psi(W)$ is continuous. Therefore the composition $T=\psi_W\circ\varphi_W^{-1}:\varphi(W)\to\psi(W)$ is continuous.
The map $T$ is bijective because it is a composition of bijections. Its inverse is $T^{-1}=\varphi_W\circ\psi_W^{-1}:\psi(W)\to\varphi(W)$. Since $\psi_W^{-1}$ and $\varphi_W$ are continuous, $T^{-1}$ is continuous. Therefore $T$ is a homeomorphism. Recalling that $W=U\cap V$, this proves that
\begin{align*}
\psi\circ\varphi^{-1}:\varphi(U\cap V)\to\psi(U\cap V)
\end{align*}
is a homeomorphism.
[/step]
