Every algebraic stack $X$ admits a presentation as quotient stack $[U/R]$ of a groupoid in algebraic spaces ($U$ is an algebraic space of objects, $R$ is an algebraic space of arrows, and there are two morphisms, source and target, from $R$ to $U$)(see Tag 04TI)
In the paper Finiteness of coherent cohomology for proper $\text{fppf}$ stacks, J. Algebraic Geom. 12 (2003), 357-366, doi:10.1090/S1056-3911-02-00321-1, by Faltings, proving the coherence of higher direct images of coherent sheaves by proper maps of algebraic stacks, the proof starts (Section 3, page 6) with the choice of a presentation $X=[U/R]$ and the remarks
1) "We may assume that $U$ is affine"
2) "Furthermore, there is an étale covering $Y\to R$ with $Y$ affine",
without any additional justification.
My question, which is probably a basic fact in the theory of algebraic stacks, is: how to justify 1) and 2)?
Even if $X$ is an ordinary scheme, this is not completely obvious because the "obvious presentation" of $X$ will not satisfy 1) if $X$ is not affine. For $X$ a quasiprojective scheme, it seems to me that it follows from the Jouanolou's trick (see e.g. The Jouanolou trick ), but it is unclear to me how to treat the general case.
Maybe in general, given a presentation $[U/R]$, it is enough to replace $U$ by a disjoint union of affine schemes covering $U$ so that 1) is satisfied, but then 2) is still unclear.
