  Let $X$ be a smooth scheme over a finite field of characteristic $p$. In
answer to a conjecture of Deligne, we establish that for any prime $\ell \neq
p$, an $\ell$-adic Weil sheaf on $X$ which is algebraic (or irreducible with
finite determinant) admits a crystalline companion in the category of
overconvergent $F$-isocrystals, for which the Frobenius characteristic
polynomials agree at all closed points (with respect to some fixed
identification of the algebraic closures of $\mathbb{Q}$ within fixed algebraic
closures of $\mathbb{Q}_\ell$ and $\mathbb{Q}_p$). The argument depends heavily
on the free passage between $\ell$-adic and $p$-adic coefficients for curves
provided by the Langlands correspondence for $\mathrm{GL}_n$ over global
function fields (work of L. Lafforgue and T. Abe), and on the construction of
Drinfeld (plus adaptations by Abe-Esnault and Kedlaya) giving rise to \'etale
companions of overconvergent $F$-isocrystals. As corollaries, we transfer a
number of statements from crystalline to \'etale coefficient objects, including
properties of the Newton polygon stratification (results of Grothendieck-Katz
and de Jong-Oort-Yang) and Wan's theorem (previously Dwork's conjecture) on
$p$-adic meromorphicity of unit-root $L$-functions.
