Let $f : \mathcal{F} \to \mathcal{G}$ be a morphism of sheaves of (complex) vector spaces on a space $X$, so $f$ is a collection of $f_U$ defined on open $U \subset X$. The image of $f$ is defined in my lectures as as $$\mathcal{I}(U) := \{a \in \mathcal{G}(U) : \text{ there exists an open cover } U_i \text{ of } U \text{ s.t. } a\vert_{U_i} \in f(\mathcal{F}(U_i))\}.$$ I want to show that this is a sheaf, but I am struggling to work with the definition, even with defining what the restriction maps $\rho_{U,V}$ should do. I have seen the fact that the category of sheaves of $\mathbb{C}$-vector spaces is abelian, and in Vakil's notes he says that
We needn't worry about restriction maps - they "come along for the ride".
But when making things concrete, it is hard for me to resolve. What am I misunderstanding here? I am fairly new to category theory and sheaves, so I would prefer something that uses as little of extra abstract machinery as possible.