Quoting from Wikipedia:

If $Y$ is a quasi-compact separated scheme and $f:X\to Y$ is a separated, quasi-finite, finitely presented morphism then there is a factorization into $X\to Z\to Y$, where the first map is an open immersion and the second one is finite.

What is the strongest statement one can make if $Y$ is only assumed to be semi-separated, i.e. $Y$ has an affine diagonal (but $f$ is still a separated morphism)?

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    $\begingroup$ You never trust Wikipedia. There is literally a stronger statement (no finite presentation assumption) for $Y$ having merely quasi-compact diagonal: stacks.math.columbia.edu/tag/05K0 $\endgroup$ – user138661 May 1 at 12:55

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