Let $h:X \to Y$ be a smooth map between manifolds with boundaries.
How does one characterize a regular value for $h$ ? I am sifting through Milnor's Topology from the Differentiable Viewpoint and nowhere does he address the subtleties -if there are any- regarding such regular values (admittedly this isn't the book's focus...), as opposed to the definition given in §2. A motivating example would be the following (c.f. §4) :
Let $f,g : M \to N$ be smooth maps between boundaryless manifolds of the same dimension, where $M$ is compact and $N$ is connected. Let $F:M \times [0,1] \to N$ be a smooth homotopy between $f$ and $g$ (here $M \times [0,1]$ is a smooth manifold with boundary $(M \times 0) \cup (M \times 1)$). I would like to show that if $y\in N$ is a regular value for $F$, then it is a regular value for $f$ and $g$.
My initial approach was to simply compute, after choosing some coordinates in $M$, the jacobian matrices of these functions, and reason with linear algebra. However the approach seemed flawed, which is why I am wondering if regular values for maps like $F$, and more generally for maps like $h$, behaved differently than the usual case of a map between boundaryless manifolds.
(Edit : This type of result is probably what I am looking for)