Higher Topos Theory Theorem 2.2.5.3

Question

The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.

We have a trivial Kan fibration of simplicial sets $p : S \rightarrow T$ where $T$ is an $\infty$-category. We wish to show that for any two vertices $x,y$ of $S$, the induced map of simplicial sets $$Map_{\mathfrak{C}[S]}(x,y) \rightarrow Map_{\mathfrak{C}[T]}(p(x),p(y))$$ is a Kan weak equivalence. We have two results, the first one is that the map $$\lvert Hom^R_T(p(x),p(y)) \rvert_{Q^\bullet} \rightarrow Map_{\mathfrak{C}[T]}(p(x),p(y))$$ where $Q^\bullet$ is the cosimplicial object defined in 2.2.2 is a Kan weak equivalence. The second one is that for any simplicial set, the map $$\pi_X : \lvert X \rvert_{Q^\bullet} \rightarrow X$$ also defined in Section 2.2.2 is also a Kan weak equivalence. Lurie says that thanks to those two results, it is enough to show that the map $$Hom^R_S(x,y) \rightarrow Hom^R_T(p(x),p(y))$$ is a Kan weak equivalence.

I was thinking of fitting all this into a commutative diagram and using the two-out-of-three property but I am struggling with it. Is this the right way to look at it or am I missing something?

If someone needs more definitions I'll gladly add them.

Answer 1

From naturality we have the following commutative diagram:

$\require{AMScd} \begin{CD} Map_{\mathfrak C[S]}(x,y) @<\sim<< |Hom^R_S(x,y)|_{Q_\bullet} @>\sim>> Hom^R_S(x,y) \\ @VVV @VVV @VVV\\ Map_{\mathfrak C[T]}(px,py) @<\sim<< |Hom^R_T(px,py)|_{Q_\bullet} @>\sim>> Hom^R_T(px,py) \end{CD}$

From the above two results the horizontal maps are weak equivalences. So if the right hand vertical map is an equivalence, then by 2/3, so is the middle vertical map, so by 2/3, so is the left hand vertical map.