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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, J)$ be a small [[site|NYE0Z94MME]] and let $P$ be a [[presheaf|AND2WHZC8P]] of sets on $\\mathcal{C}$.\nA *sheafification* of $P$ is a [[sheaf|RT8A8FJ566]] $aP$ for $J$ together with a morphism of presheaves $\\eta \\colon P \\to aP$ with this property.\nFor every sheaf $F$ and every morphism $\\varphi \\colon P \\to F$, there is exactly one morphism of sheaves $\\psi \\colon aP \\to F$ with $\\psi \\circ \\eta = \\varphi$.\n\n## Remarks\n\nA sheafification exists for every presheaf on a small site and is unique up to unique isomorphism.\nSo $P \\mapsto aP$ is a functor $\\mathrm{PSh}(\\mathcal{C}) \\to \\mathrm{Sh}(\\mathcal{C}, J)$, [[left adjoint|TZHF6ZM92S]] to the inclusion, and it preserves finite limits.\nOne construction is $aP = (P^+)^+$, where $P^+(c)$ is the colimit over covering sieves $S \\in J(c)$ of the sets of matching families for $S$.\nIf $P$ is already a sheaf, then $\\eta$ is an isomorphism.\nExample: on the site of open sets of a space $X$, the sheafification of the constant presheaf $U \\mapsto A$ is the sheaf of locally constant functions $U \\to A$.\nNon-example: let $A = \\set{0, 1}$.\nThe constant presheaf itself, with $\\eta = \\id$, is not a sheafification, since it is not a sheaf.\nIndeed, the empty sieve covers $\\emptyset$, and its unique (empty) matching family has two amalgamations in $P(\\emptyset) = A$.\nThe sheafification repairs this: the sheaf of locally constant functions has exactly one section over $\\emptyset$.\n",
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