Let $B$ be a weakly contractible category (that is, it has a weakly contractible nerve), and let $F:E\to B$ be a Grothendieck fibration. Suppose further that the ordinary fibers of the Grothendieck fibration, $F^{-1}(b)$ for each $b\in \mathrm{Ob}(B)$ are themselves weakly contractible. Does it follow that $E$ is weakly contractible?



A Grothendieck fibration over a weakly contractible category with weakly contractible fibers is weakly contractible?
This follows easily from proposition 2.1.10 of [La théorie de l'homotopie de Grothendieck](http://www.math.jussieu.fr/~maltsin/ps/prstnew.pdf) (Astérisque, 301) by G. Maltsiniotis.


The statement there is that given any functor $u:A\to B$ in $\mathrm{Cat}/C$ between (Grothendieck) prefibrations $A\to C$ and $B\to C$ whose fibers $u\times\_C c:A\times\_C c \to B\times\_C c$ for all $c\in \mathrm{Ob}(C)$ are $W$-equivalences with respect to a fixed fundamental localizer $W$ on $Cat$, the functor $u$ is a colocal $W$-equivalence over $C$ (and therefore a $W$-equivalence). 


The case in my question then follows by letting $W$ be the minimal fundamental localizer, by letting $C=B$, and letting the prefibration $B\to C$ be the identity map. 


