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Let $X$ be a topological space and $\mathcal{F}$ and $\mathcal{G}$ two sheaves of abelian groups. Now let me define a presheaf $\mathcal{F} \otimes_{\mathbb{Z}} \mathcal{G}$ such that $\mathcal{F} \otimes_{\mathbb{Z}} \mathcal{G} (U) = \mathcal{F}(U) \otimes_{\mathbb{Z}} \mathcal{G}(U) $.

Is the presheaf $\mathcal{F} \otimes_{\mathbb{Z}} \mathcal{G}$ actually a sheaf?

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No, in general this is not true.

Hint: Consider the case $\mathcal F=\mathcal G=\mathbb Z_X$ (the constant sheaf) on the discrete two point space $X=\mathrm{pt}\sqcup \mathrm{pt}$.

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