Let $N,M$ be smooth manifolds with or without boundary. Lee's ISM defines a smooth homotopy $H : N \times I \to M$ as a homotopy that extends to a smooth map in some neighborhood of $N \times I \subset N \times \mathbb{R}.$ Is this equivalent to requiring that $H$ be smooth, where $N \times I$ is interpreted as a manifold with boundary? My initial instinct is to say yes, but when unraveling definitions I'm having trouble making the jump from a smooth extension around each point on $\partial(N \times I)$ to a smooth extension around the entire $N \times I$.
EDIT: It just occurred to me that $N \times I$ will not be a manifold with boundary if $N$ itself has nonempty boundary, so my question will only apply to the case when $N$ has empty boundary.
EDIT 2: Here's the definition I'm working with for a smooth map on a manifold with boundary:
Let $F : M \to N$ be a map, where $M$, $N$ are manifolds with boundary. $F$ is smooth if for all $p \in M$, there exist charts $(U,\phi : U \to \tilde{U} \subset \mathbb{H}^n \text{ or } \mathbb{R}^n)$ for $M$ containing $p$ and $(V,\psi : V \to \tilde{V} \subset \mathbb{H}^n \text{ or } \mathbb{R}^n)$ such that $F(U) \subset V$ and $\psi \circ F \circ \phi^{-1} : \tilde{U} \to \tilde{V}$ is smooth as a map of Euclidean spaces. (If $\tilde{U} \subset \mathbb{H}^n$, we require that $\psi \circ F \circ \phi^{-1}$ extends to a smooth map in a neighborhood of each point in the domain.)