A parametrized Morse trajectory space is a manifold

The Morse--Smale condition says that the two immersion maps from $W^u(p)$ and $W^s(q)$ are transverse. Thus [F2] makes their fibre product a smooth manifold. Its evaluation image is precisely $W^u(p)\cap W^s(q)$, hence, by the point-marked convention, $\tilde{\mathcal{M}}(p,q)$.

First suppose $\lambda (p)<\lambda (q)$. The source dimensions in [F1] sum to $\lambda (p)+(n-\lambda (q))<n$, so [F2] makes the fibre product empty. If the fibre product is nonempty, [F2] instead gives its dimension as $\lambda (p)+(n-\lambda (q))-n=\lambda (p)-\lambda (q)\ge 0$. These cases prove the stated conditional dimension and emptiness assertions without assigning a negative dimension to an empty space.