A generic timelike product for a prescribed smooth Cauchy section

Apply [F2] with precisely its finite-density and volume-time inputs. Along every $X$-orbit, $t({\mathrm{\Phi }}_s(z))=t(z)+s$; hence neither parameter end can converge to a point of $D$. Each orbit is an inextendible timelike curve, so meets $C$ exactly once. Therefore $(s,c)\mapsto {\mathrm{\Phi }}_s(c)$ is surjective. If two such points agree, the common orbit meets $C$ at the same point; equality of $t$ then gives equality of the flow parameters. Thus the map is bijective.

Equip $\mathbb{R}\times C$ with the canonical product smooth structure of [F4]. The map in step 1.1 is smooth as the restriction of the global smooth flow. Its differential sends the parameter direction to $X$ and the three section directions to the flow image of $TC$. Since a spacelike three-plane cannot contain the timelike $X$, these four directions are linearly independent at parameter zero and remain so under the invertible flow differential. The map is thus a local diffeomorphism by [F3], and its local inverses combine into a smooth inverse because it is bijective, giving a diffeomorphism in the sense of [F4]. Compose its continuous inverse with the continuous product projection to $C$. This is a continuous surjection $D\to C$, so [F5] shows that $C$ is connected when $D$ is. No claim about a stationary flow was used.