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.



Higher Topos Theory Theorem 2.2.5.3
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.


