Let $A$ be an abelian group, and $X$ a topological space. Define the constant sheaf $\mathcal{A}$ on $X$ determined by $A$ as follows: for any open set $U \subset X$,
$\mathcal{A}(U)=$the group of continuous functions from $U$ to $A$, where $A$ is endowed with the discrete topology; with the usual restrictions we obtain a sheaf.
Moreover, the constant presheaf $\mathcal{F}$ of $A$ is defined by $\mathcal{F}(U)=A$ when $U \neq \phi$, again considering the usual restrictions we obtain a presheaf.
I am asked to show that the sheafification of $\mathcal{F}$ is $\mathcal{A}$.For this, I wanted to use the universal property of sheafification, and considered the application $\theta :\mathcal{F} \to \mathcal{A}$, $\theta(U)(a):=s_a$, where $s_a$ is the constant function on $U$, equal to $a$. I managed to show it is a morphism of sheaves. but given a sheaf $\mathcal{G}$ and a morphism $\varphi: \mathcal{F} \to \mathcal{G}$, I'm having trouble finding the unique morphism $\psi :\mathcal{A} \to \mathcal{G}$ s.t. $\psi \theta =\varphi$