$\newcommand{\op}{\mathrm{op}}\newcommand{\Un}{\mathrm{Un}}$Lurie's relative nerve functor $\mathrm{N}^{F}_{\bullet}$ is a simpler version of the unstraightening functor $$\Un_\phi:\mathrm{sPSh}(\mathcal{C}_\bullet)\to\mathsf{sSets}_{/S}$$ (or rather of its opposite; see HTT 3.2.5.1) which "unstraightens the homotopy coherence laws of a simplicial presheaf into a left fibration over $S_\bullet$" (I'm paraphrasing the nLab here)

Spelling out HTT 3.2.5.2, one sees that $\mathrm{N}^F_\bullet$ admits the following very simple description: it is the simplicial set where

a vertex of $\mathrm{N}^F_\bullet$ is a pair $(A,x)$ with

- $A$ an object of $\mathcal{C}$;
- $x$ a vertex of $F(A)$;

an edge of $\mathrm{N}^F_\bullet$ from $(A,x)$ to $(B,y)$ is a pair $(f,e)$ with

- $f:A\to B$ a morphism of $\mathcal{C}$;
- $e: F(f)(x)\to y$ an edge of $F(B)$;

a $2$-simplex of $\mathrm{N}^F_\bullet$ is a $9$-tuple $(f,g,x,y,z,e_{01},e_{12},e_{02},\sigma)$ with

- $f: A\to B$ a morphism of $\mathcal{C}$;
- $g: B\to C$ a morphism of $\mathcal{C}$;
- $x$ a vertex of $F(A)$;
- $y$ a vertex of $F(B)$;
- $z$ a vertex of $F(C)$;
- $e_{01}: F(f)(x)\to y$ an edge of $F(B)$;
- $e_{12}: F(g)(y)\to z$ an edge of $F(C)$;
- $e_{02}: F(g\circ f)(x)\to z$ an edge of $F(C)$;
- $\sigma: e_{12}\circ F(g)(e_{01})\Rightarrow e_{02}$ a $2$-simplex of $F(C)$ as in the diagram $$\require{AMScd} \require{cancel} \def\diaguparrow#1{\smash{\raise.6em\rlap{\!\!\!\!\!\!\scriptstyle #1} \lower.6em{\cancelto{}{\Space{2em}{1.7em}{0px}}}}} \begin{CD} && F(g)(y)\\ & \diaguparrow{F(g)(e_{01})} @VVe_{12}V \\ F(g\circ f)(x) @>>e_{02}> z \end{CD}$$

and so on.

Now, in HTT 3.2.5.5, Lurie introduces a functor $\mathfrak{F}_X$ that is (FWIU) supposed to be a simpler analogue of the straightening functor $\mathrm{St}_\phi$. However, Lurie does so by invoking the adjoint functor theorem, similarly to the definition of $\Un_\phi$.

I gather that nevertheless those functors admit nice constructions (as shown for $\Un_\phi$ in Rezk's notes); so, my question is:

Question: What is a concrete description of Lurie's $\mathfrak{F}_X$ functor?