= Solution
Radial null curves obey $-f\,du^2-2\,du\,dr=0$. For their unparameterized paths,
$$
\boxed{\frac{du}{dr}=0\quad\text{or}\quad\frac{du}{dr}=-\frac2f.}
$$
Inside the horizon, the first family satisfies $dt/dr=1/f$, so $dr/dt=f>0$ and is outgoing. The second satisfies $dt/dr=-1/f$ and is ingoing. Integrating gives $u=u_0$ for outgoing rays and $u=C-2a\operatorname{artanh}(r/a)$ for ingoing rays. The null paths are geodesics after suitable reparameterization; for example in the two-dimensional radial metric each null direction has covariant acceleration proportional to itself.
To show constant-$u$ rays at the requested angle, the diagram uses vertical plotting coordinate $u+r$ and horizontal $r$, with equal scale. This is the stated tilted $u$ grid: constant $u$ becomes a line of slope one. The ingoing curves bend to decreasing $u+r$ as $r$ approaches the horizon.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-4-de-sitter-rays.png]
{title=Outgoing and ingoing radial null rays in de Sitter space on a tilted retarded-time grid}
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