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Let $X$ be a scheme and $\mathcal F$ an $\mathcal O_X$-module. We assign an $\mathcal O_X(U)$-module to each open set $U$ of $X$, that is, $U\mapsto \mathcal F(U)\otimes_{\mathcal O_X(U)}\mathcal F(U)$, is it a presheaf or sheaf?

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    It's a presheaf, see : https://math.stackexchange.com/questions/1374056/tensor-product-of-mathscro-x-modules-which-results-in-a-presheaf – Nicolas Hemelsoet Jun 18 '19 at 09:12
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    It is not a sheaf in general, see https://math.stackexchange.com/questions/1488296/tensor-product-of-sheaves-is-not-a-sheaf – Watson Jun 18 '19 at 16:58
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    See also https://mathoverflow.net/questions/105232/tensor-product-of-sheaves-separated – Watson Jun 18 '19 at 17:03

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