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 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.