# Sheaves of abelian groups and tensor product

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?

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