Attaching two manifolds along their boundary I have a question about a proof in John Lee's Introduction to Topological Manifolds.
Suppose $M$ and $N$ are two topological $n$-manifolds with nonempty boundary (for reference, the definition I am using is: $M$ and $N$ are second countable Hausdorff spaces such that each point has an open neighborhood homeomorphic to either an open set in $\mathbf{R}^n$ or an open set in $\mathbf{H}^n = \{ (x_1,\ldots,x_n) \in \mathbf{R}^n: x_n \geq 0\}$). If $\partial M$ and $\partial N$ denote the manifold boundaries of $M$ and $N$, suppose $h: \partial N \to \partial M$ is a homeomorphism. We may attach $M$ and $N$ along their boundaries by forming the adjuction space
$$ M \cup_h N = \frac{M \sqcup N}{\sim}$$
where $\sim$ is the relation generated by identifying $x\in \partial N$ with $h(x)$. The theorem is:

With $M$, $N$, and $h$ as above, $M \cup_h N$ is an $n$-manifold (without boundary).

The proof first proceeds to show that each point of the adjuction space has a euclidean neighborhood. If $q: M \sqcup N \to M \cup_h N$ denotes the quotient map, then Lee puts $S=q(\operatorname{Int} M \cup \operatorname{Int} N)$ (where $\operatorname{Int}$ denotes the manifold interior) and shows that each point of $M \cup_h N -S$ has a euclidean neighborhood. I am OK with this argument.
Now, I quote:

Suppose $s\in S$, and let $y \in \partial N$ and $x=h(y) \in \partial M$ be two points in the fiber $q^{-1}(s)$. We can choose coordinate charts $(U,\phi)$ for $M$ and $(V, \psi)$ for $N$ such that $x\in U$ and $y\in V$, and let $\hat{U}=\phi(U)$, $\hat{V} = \psi(V) \subseteq \mathbf{H}^n$. It is useful in this proof to identify $\mathbf{H}^n$ with $\mathbf{R}^{n-1}\times [0,\infty)$ and $\mathbf{R}^n$ with $\mathbf{R}^{n-1} \times \mathbf{R}$. By shrinking $U$ and $V$ if necessary, we may assume $h(V \cap \partial N) = U \cap \partial M$, and that $\hat{U} = U_0 \times [0,\epsilon)$ and $\hat{V} = V_0 \times [0,\epsilon)$ for some $\epsilon>0$ and some open subsets $U_0,V_0 \subseteq \mathbf{R}^{n-1}$.

My question: How does one choose the coordinate domains $U$ and $V$ to satisfy these two conditions simultaneously? I know how to choose $U$ and $V$ such that $h(V \cap \partial N) = U \cap \partial M$, and I know how to choose $U$ and $V$ such that their images under the coordinate maps are equal to sets of the form $U_0 \times [0,\epsilon)$ and $V_0 \times [0,\epsilon)$, but I don't know how to pick $U$ and $V$ such that both of these conditions are satisfied at the same time.
EDIT: Let me be a bit more specific. We are to choose the coordinate domains (by shrinking if necessary) to satsify the following two conditions simultaneously:
(a) $h(V \cap \partial N) = U \cap \partial M$,
(b) $\hat{U} = U_0 \times [0,\epsilon)$ and $\hat{V} = V_0 \times [0,\epsilon)$.
Now, assuming that I have adjusted $U$ and $V$ to satisfy (a), then I may adjust them further to satisfy (b). But then how do I know that $U$ and $V$ still satisfy (a)? I can be more specific by what I mean by "adjust" if neeeded.
Am I misreading the proof, or am I missing something glaringly obvious?
 A: EDIT 6/17/19: Another reader recently pointed out that the argument I sketched below is not appropriate for this place in the book, since it uses properties of compactness that are not introduced until Chapter 4. I'll leave my original answer here for those who are curious; but a much better solution is to just abandon the idea of arranging for the images of the coordinate maps to be product open sets, because that was not really needed anyway. I've added a correction to my online list. 
Original Answer:
Here's a way to see it.  Choose charts $(U,\varphi)$ and $(V,\psi)$ such that $x\in U$ and $y\in V$, and let  $U_1 = U\cap \partial M$ (which is a  neighborhood of $x$ in $\partial M$) and   $V_1 = V\cap \partial N$ (a neighborhood of $y$ in $\partial N$).  After shrinking $U$ if necessary, we may assume that $h(U_1)\subset V_1$. (Because $U_1\cap h^{-1}(V_1)$ is a neighborhood of $x$ in the subspace topology $\partial M$ inherits from $M$, there is an open subset $\widetilde U\subset M$ such that $U_1\cap h^{-1}(V_1) = \widetilde U\cap\partial M$.  Replacing $U$ by $U\cap \widetilde U$ does the trick.)   Let $U_0\subset U_1$ be a neighborhood of $x$ that's precompact in $U_1$, and let $V_0 = h(U_0)\subset V_1$, a precompact neighborhood of $y$ in $V_1$. 
Now let $\widehat U=\varphi(U)$, $\widehat V=\psi(V)$, considered as subsets of $\mathbb H^n  = \mathbb R^{n-1}\times [0,\infty)$.  Since $\varphi(\overline {U_0})$ is a compact subset of $\mathbb R^{n-1}\times \{0\}$,  there is some $\varepsilon_1>0$ such that $\varphi(\overline {U_0})\times[0,\varepsilon_1)\subset \widehat U$.  Similarly, there's $\varepsilon_2>0$ such that $\psi(\overline {V_0})\times[0,\varepsilon_2)\subset \widehat V$.  Now just take $\varepsilon = \min(\varepsilon_1,\varepsilon_2)$, and replace $U$ by $\varphi^{-1}\big(\varphi(U_0)\times[0,\varepsilon)\big)$ and $V$ by $\psi^{-1}\big(\psi(V_0)\times[0,\varepsilon)\big)$.  
A: I take it you're worried about having the $\epsilon$'s be the same? You can rescale by a linear map in the $x_n$ coordinate.
OK, if the first condition no longer holds, shrink either $U_0$ or $V_0$ to make it hold.
