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We work with abelian sheaves on a topological space $X$. (You can assume that $X$ is "nice", locally compact, etc ...)

First, if $F$ and $G$ are two abelian sheaves, we have a natural map $$\mathcal{H}om(F,G)\otimes F \to G.$$ Indeed $$\text{Hom}(\mathcal{H}om(F,G)\otimes F,G) \simeq \text{Hom}(\mathcal{H}om(F,G),\mathcal{H}om(F,G))$$ so the identity on the RHS defines the desired morphism on the LHS. Now take $H$ an abelian sheaf and tensors the previous map by $H$. We get $$\mathcal{H}om(F,G)\otimes F \otimes H \to G \otimes H.$$ Hence we get the map $$\mathcal{H}om(F,G)\otimes H \to \mathcal{H}om(F,G\otimes H).$$ Is this last map an epimorphism ? If not, could it be one in the derived category ?

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Consider the one point space, so that abelian sheaves are just abelian groups.

$$ \hom(\mathbb{Q}, \mathbb{Z}) \otimes \mathbb{Q} \to \hom(\mathbb{Q}, \mathbb{Z} \otimes \mathbb{Q}) $$

is not an epimorphism.

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