Need help understanding comment in Higher Topos Theory

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.

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Suppose the coindtions 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 $$\mathcal{C}_{/e} \rightarrow \mathcal{D}_{/\overline{e}} \times_{\mathcal{D}_{/p(x)}} \mathcal{C}$$ 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.