In section 2.4.4 of Lurie's Higher Topos Theory, it is said multiple times that using

Proposition: Let $p : \mathcal{C} \rightarrow \mathcal{D}$ be an inner fibration of $\infty$-categories. Let $x,y$ be vertices of $\mathcal{C}$, let $\tilde{e} : p(x) \rightarrow p(y)$ be an edge of $\mathcal{D}$ and let $ e: x'\rightarrow y$ be a locally $p$-Cartesian edge of $\mathcal{C}$ lifting $\tilde{f}$. Then in the homotopy category $\mathcal{H}$ of spaces there is a fiber sequence $$ Map_{\mathcal{C}_{p(x)}}(x,x') \rightarrow Map_{\mathcal{C}}(x,y) \rightarrow Map_{\mathcal{D}}(p(x),p(y))$$ where the fiber is taken over $\tilde{e}$.

we have that any cartesian fibration of simplicial set $p :\mathcal{C} \rightarrow \mathcal{D}$ is fully faithful map in the sense that for every vertices $x,y \in \mathcal{C}$, the induced map $$Map_{\mathcal{C}}(x,y) \rightarrow Map_{\mathcal{D}}(p(x),p(y))$$ is a Kan weak equivalence of simplicial sets.

I see that from the fact that $p$ is a Cartesian fibration we have that $p$ satisfies the hypothesis of the above proposition. However I do not see how we can say that $p$ is fully faithful from it.