  Berthelot's conjecture predicts that under a proper and smooth morphism of
schemes in characteristic $p$, the higher direct images of an overconvergent
$F$-isocrystal are overconvergent $F$-isocrystals. In this paper we prove that
this is true for crystals up to isogeny. As an application we prove a K\"unneth
formula for the crystalline fundamental group.
