Lurie claims the following is a formal consequence.
[2.4.1.3] Let $p:X \rightarrow S$ be an isomorphism of simplicials. Then every edge of $X$ is $p$-cartesian.
Definition of $p$-Cartesian is given at 2.4.1.1.
I would appreciate if someone can outline what is done to prove this. I am still very unfamiliar with the language of simplicial sets.
I will use the notation in Lurie's. We have the isomorphisms in $Set_{\Delta}$, $$ X_{/y} \cong S_{/p(y)} \quad \quad X_{/f} \cong S_{/p(f)} $$
And also the simple fact: for objects $a,b$ in a category, $a \times_a b \cong b$, where we take the pullback with respect to a morphism $b \rightarrow a$ and the identity $a \rightarrow a$.