Firstly, a bit of notation. Let $C$ be a simplicial set. We define, for $x,y \in C$ vertices in $C$

$$Map(x,y) = \{x\}\times_{\Delta^{\{0\}}}Map_{sSet}(\Delta^1,C) \times_{\Delta^{\{1\}}} \{y\} $$

the sSet of arrows from x to y. You can imagine a $n$ simplex here as a cylinder with basis $\Delta^n$, with all $1_x$ on the left base and all $1_y$ on the right base. From this description it is evident that $Map(x,y) \simeq (C_{/y})_x$, the cones over y with all $1_x$ at the basis. Symmetrically, it is equivalent to $(C_{x/})_y$.

As the two sSet are fibers of a right and a left fibration, they are respectively left inner fibrant and right inner fibrant. Thus, the map sets are Kan complexes.

I am searching to generalize this to an arbitrary inner fibration $p:C \to D$, and show that $Map(x,y) \to Map(px,py)$ is a Kan fibration.

Thanks!