Let $X$ be a topological space and $V\subseteq U\subseteq X$ open sets. Let $\mathcal{F}, \mathcal{G}$ be topological sheaves of abelian groups (or other structures). Let $\alpha: \mathcal{F}\rightarrow \mathcal{G}$ be a morphism of sheaves. I understand why $\ker(\alpha)$ is a sheave, but I still have no intuition why this is not necessarily true for $\text{Im}(\alpha)$.
Is there an instructive example for $\text{Im}(\alpha)$ not being a sheave? To what extent does this property depend on the codomain category of $\mathcal{F}, \mathcal{G}$?