  A Lorentzian manifold endowed with a time function, $\tau$, can be converted
into a metric space using the null distance, $\hat{d}_\tau$, defined by Sormani
and Vega. We show that if the time function is a proper regular cosmological
time function as studied by Andersson, Galloway and Howard, and also by Wald
and Yip, or if, more generally, it satisfies the anti-Lipschitz condition of
Chru\'sciel, Grant and Minguzzi, then the causal structure is encoded by the
null distance in the following sense: $$ \hat{d}_\tau(p,q)=\tau(q)-\tau(p) \iff
q \textrm{ lies in the causal future of } p. $$ As a consequence, in dimension
$n+1$, $n\ge 2$, we prove that if there is a bijective map between two such
spacetimes, $F: M_1\to M_2$, which preserves the cosmological time function, $
\tau_2(F(p))= \tau_1(p)$ for any $ p \in M_1$, and preserves the null distance,
$\hat{d}_{\tau_2}(F(p),F(q))=\hat{d}_{\tau_1}(p,q)$ for any $p,q\in M_1$, then
there is a Lorentzian isometry between them, $F_*g_1=g_2$. This yields a
canonical procedure allowing us to convert such spacetimes into unique metric
spaces with causal structures and time functions. This will be applied in our
upcoming work to define Spacetime Intrinsic Flat Convergence.
