Consider the following construction: Define $G_n$ to be the contractible groupoid on $n+1$ objects. Choosing a linear order on the objects of each $G_n$ turns $G_*$ into a cosimplicial object. Define the cosimplicial simplicial set $J_*=N(G_*)$.

This cosimplicial object defines a Quillen pair between the Quillen model structure on sSet and the Joyal model structure. The right adjoint is the "core Kan complex" functor.

Given a pair of simplicial sets $X,Y$, define $\operatorname{Map}(X,Y)_n=\operatorname{Hom}(J_n \times X,Y)$.

Then when $X$ is a quasicategory and $x,x'\in X_0$ are vertices, define the Rezk hom $$X(x,x')=\operatorname{Map}(\Delta[1],X)\times_{\operatorname{Map}(\partial\Delta[1],X)} \Delta[0]$$ to be the fibre over $(x,x')$.

This definition is actually just the definition of the Hom of the associated complete Segal space.

Question: When $X$ is the coherent nerve of a fibrant simplicial category, $C$, has anyone computed a direct comparison between the Rezk Hom and the Hom of $C$.

Lurie compares a different presentation of this hom object (the right hom) using straightening/unstraightening over a point.

What we can do is define $$K_n=\partial\Delta[1] \coprod_{J_n\times\partial\Delta[1]} J_n \times \Delta[1]$$ and define $Q_n=\operatorname {Hom}_{\mathfrak{C}[K_n]}(0,1)$, but the combinatorics looks really crazy to show that $Q_n$ is naturally weakly equivalent to the $n$-simplex.

Has anyone worked this out without using straightening/unstraightening? Can we simplify the $J_i$ by replacing them all with finite fragments the way people usually do with $J_1$?