  This article is the second one of a series of three articles devoted to
direct images of isocrystals: here we consider convergent isocrystals with
Frobenius structure. Let V be a complete discrete valuation ring, with residue
field k = V/m of characteristic p > 0 and fraction field K of characteristic 0.
Firstly we characterize convergent F-isocrystals on a smooth affine k-scheme.
Secondly, for perfect k and after a detailed exposition of the Teichm\^uller
liftings, especially for the affine rigid line, we derive the existence of
Frobenius isomorphisms on the direct images of convergent F-isocrystals under a
proper smooth and liftable k-morphism.
