  Let X be a smooth variety over a field of characteristic p>0. We prove that
the forgetful functor from the category of overconvergent F-isocrystals on X to
the category of convergent F-isocrystals is fully faithful. The argument uses
the quasi-unipotence theorem for overconvergent F-isocrystals (recently proved
independently by Andre, Mebkhout, and the author; see math.AG/0110124), plus
arguments of de Jong. In the process, we establish a theorem of Quillen-Suslin
type (i.e., every finite projective module is free) over rings of
overconvergent power series.
