Lemma 6.19.1. 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:

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 $\mathcal{F}$.

For any morphism $\mathcal{F} \to \mathcal{G}$, where $\mathcal{G}$ is a sheaf with values in $\mathcal{C}$ there exists a unique factorization $\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G}$.

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