In Remark 2.4.1.9 in Lurie's Higher Topos Theory, he asserts that if $p:X\to S$ is an inner fibration, x a vertex of X, and $\overline{f}: \overline{x}'\to p(x)$ an edge of S ending at $p(x)$, then $X_{/x}\times_{S_{/p(x)}} \{\overline{f}\}$ is an $\infty$-category. I tried to convince myself of this by showing the induced map $\pi: X_{/x}\to S_{/p(x)}$ was an inner fibration, as it would then follow, by stability under pullback, that the left vertical arrow in the diagram below is an inner fibration. $\require{AMScd}$ \begin{CD} X_{/x}\times_{S_{/p(x)}} \{\overline{f}\} @>{}>> X_{/x}\\ @VVV @VVV\\ \Delta^0 @>{\overline{f}}>> S_{/p(x)} \end{CD}
However, I'm not actually sure why/if this is true. Moreover, is it true that an edge $f: x' \to x$ is p-Cartesian if and only if it is a final object in the fibre product $X_{/x}\times_{S_{/p(x)}} \{\overline{f}\}$? In this Remark, the forward direction is claimed, and I think I kind of see why, but is the backwards direction true as well?
If anyone has insight into this, in particular, if it is true that inner (or even left/right) fibrations are preserved under formation of over/under categories, I would really appreciate it!