Lemma 56.5.6. In Lemma 56.5.2 or in Lemma 56.5.5 if $F$ is an exact functor, then the corresponding object $\mathcal{K}$ of $\mathit{QCoh}(\mathcal{O}_{X \times _ R Y})$ is flat over $X$.

**Proof.**
We may assume $X$ is affine, so we are in the case of Lemma 56.5.2. By Lemma 56.5.4 we may assume $Y$ is affine. In the affine case the statement translates into Remark 56.3.5.
$\square$

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