Quasi-coherence of direct image under a quasi-compact quasi-separated morphism

If $f:X\to Y$ is quasi-compact and quasi-separated, then the direct image of every quasi-coherent sheaf on $X$ is again a quasi-coherent sheaf. A finite affine cover of each inverse image of an affine target chart expresses the direct image as the kernel of a morphism between finite sums of quasi-coherent direct images from affine charts.