# Is the canonical morphism $\operatorname{Pre}(\operatorname{im}\theta) \to \operatorname{im}\theta$ injective?

Let $$X$$ be a topological space. Let $$\mathcal F, \mathcal G$$ be two sheaves of Abelian groups on $$X$$ and $$\theta: \mathcal F\to \mathcal G$$ a morphism. Denote the presheaf of the image by $$\operatorname{Pre}(\operatorname{im}\theta)$$. Is the canonical morphism $$\operatorname{Pre}(\operatorname{im}\theta) \to \operatorname{im}\theta$$ injective?