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

Question

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?