I am studying orientations on manifolds in chapter 15 of Lee's Introduction to Smooth Manifolds, and I'm having trouble with a small technical detail in the statement of lemma 15.27. The lemma reads as follows:
Let $M$ be an oriented smooth $n$-manifold with boundary. Suppose $U \subseteq \mathbb{R}^{n-1}$ is open, $a, b$ are real numbers with $a < b$, and $F : (a,b] \times U \to M$ is a smooth embedding that restricts to an embedding of $\{b\} \times U$ into $\partial M$. Then the parametrization $f : U \to \partial M$ given by $f(x) = F(b, x)$ is orientation-preserving for $\partial M$ if and only if $F$ is orientation-preserving for $M$.
For a little context, the orientation on $\partial M$ is that given by proposition 15.24, and the orientations on $(a, b] \times U, U$ are those determined by proposition 15.11 from the standard orientations on $\mathbb{R}^n, \mathbb{R}^{n-1},$ respectively. (The standard orientation on $\mathbb{R}^n$ is the one determined by the standard coordinate frame if $n > 0$, and is the orientation obtained by giving $0$ the orientation $+1$ if $n = 0$.)
My issue is that by definition (p. 383 in Lee's book), an orientation-preserving map must be a local diffeomorphism. While it is clear that $f$ is always a local diffeomorphism, it is not clear to me that $F$ is always a local diffeomorphism, since we cannot apply proposition 4.8 because of the nonempty boundary of $(a, b] \times U.$ So, to summarize, my question is:
For $a, b \in \mathbb{R}$ with $a < b$, an open subset $U \subseteq \mathbb{R}^{n-1}$, and a smooth $n$-manifold $M$ with boundary, let $F : (a, b] \times U \to M$ be a smooth embedding that restricts to an embedding of $\{b\} \times U$ into $\partial M$. Is $F$ necessarily a local diffeomorphism? What about if $M$ is orientable? What about if for some orientation on $M$, $f : U \to \partial M$ as defined above is orientation-preserving?
