Remark 8.6.6.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let
be the left fibration of Proposition 8.1.1.15. If $X$ and $Y$ are vertices of $\operatorname{\mathcal{E}}$ and $f: U(X) \rightarrow U(Y)$ is an edge of $\operatorname{\mathcal{C}}$, then the fiber $\lambda ^{-1} \{ (X,f,Y) \} $ is a Kan complex, which is homotopy equivalent to the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{f} }( X, Y )$, where we abuse notation by identifying $X$ and $Y$ with their preimages in the $\infty $-category $\operatorname{\mathcal{E}}_{f} = \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. Consequently, if $\mathscr {H}$ is a relative $\operatorname{Hom}$-functor for $U$, then we have homotopy equivalences $\mathscr {H}(X,f,Y) \simeq \operatorname{Hom}_{\operatorname{\mathcal{E}}_{f}}(X,Y)$.