I am having trouble understanding this part in *Lurie's Higher Topos Theory*. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.

Lemma 2.4.4.1.Let $p : \mathcal{C} \rightarrow \mathcal{D}$ be an inner fibration of $\infty$-categories and let $X, Y \in \mathcal{C}$. The induced map $$\phi : Hom^R_{\mathcal{C}}(X,Y) \rightarrow Hom^R_{\mathcal{C}}(p(X),p(Y))$$ is a Kan fibration.[...]

Suppose the conditions of Lemma 2.4.4.1 are satisfied. Let us consider the problem of computing the fiber of $\phi$ over a vertex $\overline{e} : p(X) \rightarrow p(Y)$ of $Hom^R_{\mathcal{C}}(p(X), p(Y))$. Suppose that there is a $p$-Cartesian edge $e : X' \rightarrow Y$ lifting $\overline{e}$. By definition, we have a trivial fibration

$$\psi : \mathcal{C}_{/e} \rightarrow \mathcal{C}_{/y} \times_{}\mathcal{D}_{/p(y)} \mathcal{D}_{/\overline{e}}.$$

Consider the $2$-simplex $\sigma = s_1(\overline{e})$ regarded as a vertex of $\mathcal{D}_{/\overline{e}}$. Passing to the fiber, we obtain a trivial fibration

$$ F \rightarrow \phi^{-1}(\overline{e}),$$ where $F$ denotes the fiber of $\mathcal{C}_{/e} \rightarrow \mathcal{D}_{/\overline{e}} \times_{\mathcal{D}} \mathcal{C}$ over the point $(\sigma, x)$.

I am not understanding exactly what he means by "taking fibers". I see that both $F$ and $\phi^{-1}(\overline{e})$ are fibers, but I don't know how he obtains the trivial fibration between them. I was thinking that both those fibers are defined by pullbacks which are actually homotopy pullbacks and maybe we can find a map of diagram in which all component are weak equivalences to have that those two fibers are weakly equivalent. However I can't find those map and it would not show the that this map is a fibration.