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    $\begingroup$ Minor note: maybe it should be "separation" instead of "separification" $\endgroup$ Commented Oct 13, 2009 at 19:08
  • $\begingroup$ I like "separification". I've also heard people propose "sheafication" instead of "sheafification". $\endgroup$ Commented Oct 18, 2009 at 2:20
  • $\begingroup$ Your proof that $U$ respects limits does not work. Here is a different approach: Since affine schemes are separated, it is enough to prove the following If $\{X \to X_i\}$ is a cone of schemes such that for all affine schemes $T$ the induced map $Hom(T,X) \to lim_i Hom(T,X_i)$ is bijective, then this is already true for all schemes $T$. But this is easy because both sides are sheaves in $T$ with respect to the Zariski Topology. $\endgroup$ Commented Jun 14, 2011 at 15:29
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    $\begingroup$ This question seems to be treated in arxiv.org/abs/1510.06588 $\endgroup$ Commented Oct 29, 2015 at 16:35