  This article is the third one of a series of three articles devoted to direct
images of isocrystals: here we consider overconvergent isocrystals with
Frobenius structure. For a liftable proper smooth morphism we establish the
overconvergence of direct images, owing to the first article and the existence
of lifts of Frobenius. This result partially answers a conjecture of Berthelot
on the overconvergence of direct images of overconvergent F-isocrystals under a
proper smooth morphism.
