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?
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This follows easily from proposition 2.1.10 of La théorie de l'homotopie de Grothendieck (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. 

