Let $F$ be a presheaf over a topological space $X$ with values in a category $\mathcal{A}$ for which sheaves are defined. The sheafification of $F$, if it exists, is a sheaf $F^{{\prime}}$ over $X$ together with a morphism $\theta:F\longrightarrow F^{{\prime}}$ satisfying the following universal property:

For any sheaf $G$ over $X$ and any morphism of presheaves $\phi:F\longrightarrow G$ over $X$, there exists a unique morphism of sheaves $\psi:F^{{\prime}}\longrightarrow G$ such that the diagram

$$\xymatrix{F\ar[r]^{{\theta}}\ar@/_{1}pc/[rr]_{{\phi}}&F^{{\prime}}\ar[r]^{{\psi}}&G}$$

commutes.

In light of the universal property, the sheafification of $F$ is uniquely defined up to canonical isomorphism whenever it exists. In the case where $\mathcal{A}$ is a concrete category (one consisting of sets and set functions), the sheafification of any presheaf $F$ can be constructed by taking $F^{{\prime}}(U)$ to be the set of all functions $s:U\longrightarrow\bigcup_{{p\in U}}F_{p}$ such that

1. 1.

   $s(p)\in F_{p}$ for all $p\in U$

2. 2.

   For all $p\in U$, there is a neighborhood $V\subset U$ of $p$ and a section $t\in F(V)$ such that, for all $q\in V$, the induced element $t_{q}\in F_{q}$ equals $s(q)$

for all open sets $U\subset X$. Here $F_{p}$ denotes the stalk of the presheaf $F$ at the point $p$.

The following quote, taken from [1], is perhaps the best explanation of sheafification to be found anywhere:

$F^{{\prime}}$ is “the best possible sheaf you can get from $F$”. It is easy to imagine how to get it: first identify things which have the same restrictions, and then add in all the things which can be patched together.