Hard as I tried, I couldn't find a proof of Remark 2.2.2.11 in Higher Topos Theory, or prove it myself. It seems to need an explicit formulation for the unstraightening functor, so my question is: is an explicit expression known for the unstraightening? Anyway, is it possible to obtain Remark 2.2.2.11 without having one?



Explicit expression of the unstraightening functor
Wow, I have always thought that unstraightening has to be easier than straightening, but I've never actually looked at Lurie's treatment before, so I'm surprised to realize he defines straightening directly while defining unstraightening as its right adjoint.


Anyway, I'm pretty sure that Remark 2.2.2.11 follows from the description of straightening over a point given between Remarks 2.2.2.5 and 2.2.2.6, by applying the functoriality properties of Proposition 2.2.1.1. After all, in Remark 2.2.2.11, we are calculating the following quantity: 


$Un\_\phi \mathcal F \times\_S \{s\} = Map\_S(\{s\} \to S, Un\_\phi \mathcal F)
= Nat\_{\mathcal C}(St\_\phi(\{s\} \to S), \mathcal F)$.


where we have $\phi: \mathfrak C[S]^\mathrm{op} \to \mathcal C$, $s \in S$, and $\mathcal F: \mathcal C \to \mathsf{sSet}$. Note that


$St\_\phi(\{s\} \to S) = St\_\phi((\{s\} \to S)\_!(\{s\}=\{s\})) \\
\qquad ~~~\qquad = St\_{\phi \circ \mathfrak C(\{s\} \to S)}(\{s\}) = (\phi(s))\_! \circ St\_{\{s\}}(\{s\})$


where in the last two steps, we have applied Prop 2.2.2.1(1) and Prop 2.2.2.1(2) respectively. We have also written $\phi(s)$ for $\phi \circ \mathfrak{C}(\{s\} \to S)$, and sometimes written $\{s\}$ for the identity map $\{s\} = \{s\}$.


Now we calculuate:


$Nat\_{\mathcal C}((\phi(s))\_! \circ St\_{\{s\}}(\{s\}),\mathcal F) = Nat\_{\{s\}}(St\_{\{s\}}(\{s\}), \mathcal F(\phi(s))) \\
\qquad \qquad \qquad \qquad \qquad = Nat\_{\{s\}}(|\{s\}|\_{Q\_\bullet}, \mathcal F(\phi(s))) = Map\_{\{s\}}(\{s\}, Sing\_{Q\_\bullet} \mathcal F(\phi(s))) \\
\qquad \qquad \qquad \qquad \qquad = Sing\_{Q\_\bullet} \mathcal F(\phi(s))$


as desired. Here we have applied adjointness, then the description of straightening over a point from between 2.2.2.5 and 2.2.2.6, then adjointness again, and finally evaluated at a point.


The upshot is that we are getting an explicit *partial* description of the unstraightening functor, in the sense that we are seeing what its fibers are.


