  Let $X$ be a smooth scheme over a finite field of characteristic $p$.
Consider the coefficient objects of locally constant rank on $X$ in $\ell$-adic
Weil cohomology: these are lisse Weil sheaves in \'etale cohomology when $\ell
\neq p$, and overconvergent $F$-isocrystals in rigid cohomology when $\ell=p$.
Using the Langlands correspondence for global function fields in both the
\'etale and crystalline settings (work of Lafforgue and Abe, respectively), one
sees that on a curve, any coefficient object in one category has "companions"
in the other categories with matching characteristic polynomials of Frobenius
at closed points. A similar statement is expected for general $X$; building on
work of Deligne, Drinfeld showed that any \'etale coefficient object has
\'etale companions. We adapt Drinfeld's method to show that any crystalline
coefficient object has \'etale companions; this has been shown independently by
Abe--Esnault. We also prove some auxiliary results relevant for the
construction of crystalline companions of \'etale coefficient objects; this
subject will be pursued in a subsequent paper.
