DStevenson Notes on classification of left fibrations

Given a left fibration $X\to S$, we want to show that there is a map $S\to \mathcal{S}$ such that $X\to S$ is induced from the universal left fibration $\mathcal{S}_{1/}\to \mathcal{S}$ in the sense that there is a homotopy pullback diagram (??in the Joyal model structure on $\mathbf{S}$??)

Furthermore, we want to show that the classifying map $S\to \mathcal{S}$ is unique up to equivalence.

Observe firstly that there are 1-1 correspondences between commutative diagrams

in $\mathbf{S}$, maps

in $\mathbf{S}$, and finally simplicial functors

Let $v$ denote the cone point in $1\star X$ and let $\mathcal{M}$ denote a fibrant replacement of $\mathfrak{C}(1\star X\cup_X S)$.

Then we obtain a simplicial functor

by the formula