Suppose I have a Morse function $f$ on a compact smooth manifold $M$, potentially with boundary, and that $h$ is an automorphism of $M$ isotopic to the identity automorphism. Then is $f\circ h$ a Morse function?
It seems clear to me that critical points of $f$ will under $h^{-1}$ be mapped to critical points of $f\circ h$ and that these points will still be locally quadratic (that is, non-degenerate.) But that these should be the only critical points is slightly mysterious to me.