Following Grothendieck, let us say that a category is **aspheric** if its nerve is weakly contractible and a functor $u : \mathcal{A} \to \mathcal{B}$ is **aspheric** if for every object $b$ in $\mathcal{B}$, the comma category $(u \downarrow b)$ is aspheric. 


**Question.** Consider a pullback diagram in $\mathbf{Cat}$,
$$\require{AMScd}
\begin{CD}
\mathcal{E} @>{v}>> \mathcal{F} \\
@V{p}VV @VV{q}V \\
\mathcal{A} @>>{u}> \mathcal{B}
\end{CD}$$
where $q : \mathcal{F} \to \mathcal{B}$ (hence also $p : \mathcal{E} \to \mathcal{A}$) is a Grothendieck fibration and $u : \mathcal{A} \to \mathcal{B}$ is aspheric. Is $v : \mathcal{E} \to \mathcal{F}$ also aspheric?




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Of course, an aspheric functor is the same thing as a homotopy coinitial functor, so Thomason's homotopy colimit theorem implies that $N (v) : N (\mathcal{E}) \to N (\mathcal{F})$ is a weak homotopy equivalence, which is a strictly weaker conclusion. 


I believe I have a proof that the answer to my question is yes, so what I am really looking for is an answer *in the literature*. I would not be surprised if it appears in *Pursuing Stacks*, but perhaps someone more familiar with the document knows for sure?



Aspheric functors and Grothendieck fibrations
You may find this in [Maltsiniotis'Astérisque 301](https://www.imj-prg.fr/~georges.maltsiniotis/ps/prstnew.pdf): it is equivalent to prove the dual version (using Proposition 1.1.22), and then, Example 3.2.2 show that this is a particular case of Corollary 3.2.13 of loc. cit. This kind of ideas is developped systematically in his paper [Structures d’asphéricité, foncteurs lisses, et fibrations](http://www.numdam.org/numdam-bin/fitem?id=AMBP_2005__12_1_1_0).


