Let $X$ be a compact manifold with boundary $Y$, and $f:X\to\mathbb R$ be a Morse function, whose gradient is tangent on boundary.
As in the book, Monopole and Three Manifold, we know that $Cri(f)=C^o\cup C^s\cup C^u$, where $C^o$ means the critical points inside, $C^s$ means boundary stable critical points, and $C^u$ means the boundary unstable critical points.
Q:
For the chain map $\bar\partial:\bar C:=C^s\oplus C^u\to \bar C$, we know there is $\bar\partial^s_u$, i.e. there is a trajectory from $C^s$ to $C^u$ along the boundary, but trajectry $\partial^u_s:C^u\to C^s$, the path must leave the boundary. I donot understand this is why?