Let $(X,F)$ be a presheaf, $(X,G)$ a sheaf, and $\phi:F\rightarrow G$ a presheaf morphism (i.e. a natural transformation. I am trying to prove that $(X,F^+)$ defined by:
$$F^+(U)\left\{(s_p)\in \prod_{p\in U}F_p: \forall p\in U, \exists V_p\subset U\text{ open, and }f\in F(V_p), s.t. f_q=s_q, \forall q\in V_p\right\}$$
satisfies the universal property that there exists a unique morphism $\psi$ such that $\psi\circ\text{sh}=\phi$. I have already shown that sheafification gives a sheaf, and that $\text{sh}: F\rightarrow F^+$ defines a natural transformation, but I do not know how to define the map $F^+\rightarrow G$.
I thought maybe I could show that $\text{sh}:G\rightarrow G^+$ is an isomorphism, and then by applying sheafification I could get my unique morphism, but I am having trouble showing that $\text{sh}:G\rightarrow G^+$ is surjective. In particular, on opens I define $\text{sh}_U(s)=(s_p)$, so given $(s_p)\in G^+(U)$ I need to find a way to get an $s\in G(U)$. I believe I have to use the gluing axiom, but I can't quite things to agree on overlaps, only on open neighborhoods of overlaps. Is there something I am missing?
Given an open set $U$ you need to define a map $F^+(U)\to G(U)$. So pick an element in $F^+(U)$. It will be a sequence $(s_p)_{p\in U}$. You need to define how this sequence is mapped to an element of $G(U)$. This is accomplished by using the presheaf morphism $F\to G$. When specialized to the open set we have a morphism of (groups?) $F(U)\to G(U)$.
For each $p\in U$, there is an open set $V_p$, containing $p$ and $V_p\subseteq U$, such that $s_q = f_q$ for some $f\in F(V_p)$ and all $q\in V_p$. Now you can map $f\in F(V_p)$ to its corresponding element in $G(V_p)$ using the morphism $F\to G$.
The above procedure can be done for any point $p\in U$. Let us call the corresponding element in $G(V_p)$ by the symbol $g(V_p)$. Now you have constructed $g(V_p) \in G(V_p)$, these open sets $V_p$ cover $U$ and the choices $g(V_p)$ will agree on overlaps. From here you can glue them together and get the desired element in $G(U)$.