In this quick post, I wish to state a simple Lemma.
Let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Let $\mathcal{F}$ be a presheaf with values in $\mathcal{C}$ on $X$. Then there exists a sheaf $\mathcal{F}^{\#}$ with values in $\mathcal{C}$ and a morphism $\mathcal{F} \to \mathcal{F}^{\#}$ of presheaves with values in $\mathcal{C}$ with the following properties:
1. The map $\mathcal{F} \to \mathcal{F}^{\#}$ identifies the underlying sheaf of sets of $\mathcal{F}^{\#}$ with the sheafification of the underlying presheaf of sets of $\mathcal{F}$.
2. For any morphism $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf with values in $\mathcal{C}$, there exists a unique factorization $\mathcal{F} \to \mathcal{F}^{\#} \to \mathcal{G}$.
Proof: We start with a pre-sheaf $\mathcal{F}$ with values in $\mathcal{C}$ (some category). However, this is a pre-sheaf of abelian groups. We can use a forgetful functor
$$F(\mathcal{F}): Open(X)^{op} \to Sets $$
Now with the classical sheafification, we get a sheaf of sets $F(\mathcal{F})^{\#}$. So, we have the elements and can be glued.
We wish to lift the algebraic structure for this sheaf of sets. We can assume that $\mathcal{C}$ has filtered colimits. These colimits preserve the algebraic operations. For any $U \in X$, $\mathcal{F}^{\#}$ is a canoncial object in $\mathcal{C}$.
The second part of the lemma, which is a universal property where $\mathcal{G}$ is a sheaf in $\mathcal{C}$
$$\mathcal{F} \to \mathcal{F}^{\#} \to \mathcal{G}$$
Let us assume a morphism $\mathcal{F} \to \mathcal{G}$, then the forgetful functor gives
$$F(\mathcal{F}) \to F(\mathcal{G})$$
and since $F(\mathcal{G})$ is a sheaf, $F(\mathcal{F})^\#$ (which is a sheafification of $F(\mathcal{F}$) is a sheaf. And as we saw that $F$ creates a colimit, there is a morphism in the category $\mathcal{C}$. There is a morphism lifted $\mathcal{F}^\# \to \mathcal{G}$ and thus one has the factorization $\mathcal{F} \to \mathcal{F}^{\#} \to \mathcal{G}.$
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