**Background**

I have some difficulty to understand remark 2.4.1.9 in Lurie's HTT, which says a $p$-Cartesian edge is determined up to equivalence.

To be more precise, let $p:X\to S$ be an inner fibration of simplicial sets, let $x$ be a vertex of $X$, and let $\bar{f}:\bar{y}\to p(x)$ be an edge of $S$ ending at $p(x)$. If there are two $p$-Cartesian edges $f:y\to x$ and $f':y'\to x$ in $X$ with $p(f)=p(f')=\bar{f}$, then we have that $f$ and $f'$ are strong final objects of the $\infty$-category $K\stackrel{\text{def}}{=}X_{/x} \times_{S{/p(x)}}\{\bar{f}\}$, and thus they are equivalent $K$.

That's true. Since for a $p$-Cartesian edge $f$,
$$q:X_{/f}\to X_{/x}\times_{S_{/p(x)}}S_{/p(f)}$$
is a trivial fibration, and $K_{/\{f\}}\to K$ is a pullback of $q$.

**My Question:**

But since Lurie's original words are *"($p$-Cartesian edge) is therefore determined up to equivalence by $\bar{f}$ and $x$"*, I think we should conversely have that an object who is equivalent with a $p$-Cartesian edge is also a $p$-Cartesian edge. And he indeed utilized such a claim in the proof of Proposition 2.4.2.4 in HTT. However I cannot figure out a detailed proof. I hope someone can help me.