I am reading Vakil's notes, Section 2.3.4.
Let $\mathscr{F},\mathscr{G}$ be presheaves of abelian groups on a topological space $X$. Let $\phi:\mathscr{F}\rightarrow \mathscr{G}$ be a morphism between presheaves. The presheaf kernel $\ker_{\text{pre}}\phi$ is defined by $\ker_{\text{pre}}\phi(U)=\ker\phi(U)$ for each open subset $U\subseteq X$. Show that $\ker_{\text{pre}}\phi$ is a presheaf.
Now we define the restriction map when $U\hookrightarrow V$. I can show that the following diagram commute:
as long as all the maps in the diagram are group homomorphisms.
My question:
Are all the maps in the diagram assumed to be group homomorphism? In the notes they are defined as maps. So I am not sure. More in general, are all the maps in this section (about presheaf and sheaf) morphisms in their corresponding category? (For example, module homomorphism for mudules, ring homomorphisms for rings, etc.)