  Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring,
$k$ its residual field, $\mathcal{P}$ a proper smooth formal scheme over
$\mathcal{V}$, $P$ its special fiber, $T$ a divisor of $P$, $U:=P\setminus T$,
$Y$ a smooth closed subscheme of $U$. We prove that the category of
overconvergent $F$-isocrystals on $Y$ is equivalent to the category of
overcoherent $F$-isocrystals on $Y$. More generally, we prove such an
equivalence by gluing for any smooth variety $Y$ over $k$. Moreover, we check
that overcoherent $F$-complexes of arithmetic $\mathcal{D}$-modules split in
overconvergent $F$-isocrystals.
