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  "body": "\n## Statement\n\nLet $\\mathcal{C}$ be a small [[category|A2DE66937D]], let $\\widehat{\\mathcal{C}} = [\\mathcal{C}^{\\mathrm{op}}, \\mathbf{Set}]$ be the [[functor category|1X0WKFQX8K]] of [[presheaves|AND2WHZC8P]] on $\\mathcal{C}$, and let $y \\colon \\mathcal{C} \\to \\widehat{\\mathcal{C}}$ be the [[Yoneda embedding|4DKB6AQ8J7]].\nLet $P$ be a presheaf on $\\mathcal{C}$, and let $\\int P$ be its [[category of elements|S9EJKKRHW1]], with objects the pairs $(c, x)$ such that $x \\in P(c)$.\nLet $\\pi \\colon \\int P \\to \\mathcal{C}$ be the projection $(c, x) \\mapsto c$.\nThen $P$ is the [[colimit|BJS1CGTKQ5]] of the diagram $y \\circ \\pi \\colon \\int P \\to \\widehat{\\mathcal{C}}$.\nThe colimit cocone has components $y(c) \\to P$ given by the natural transformations that correspond to $x \\in P(c)$ under the Yoneda lemma.\n\n## Remarks\n\nProof idea: colimits in $\\widehat{\\mathcal{C}}$ are computed objectwise in $\\mathbf{Set}$.\nAt an object $d$, the colimit of the sets $\\Hom(d, c)$ over $(c, x) \\in \\int P$ is a quotient of the set of pairs $(g \\colon d \\to c, x \\in P(c))$, and each pair is identified with $(\\id_d, P(g)(x))$.\nSo the colimit at $d$ is $P(d)$, and the identifications are compatible with the maps of $P$.\n\nExample: if $P = y(a)$ is itself [[representable|KCCHH87H6E]], then $\\int P$ is the slice category over $a$, which has the terminal object $(a, \\id_a)$, and the colimit is just $y(a)$.\n\nIn short: every presheaf is a colimit of representable presheaves.\nSmallness of $\\mathcal{C}$ makes $\\int P$ small, so the colimit is a small colimit.\n\n## Explanations\n\n### L1 human\n\nRepresentable presheaves are the building blocks of all presheaves.\nEvery presheaf can be glued together from copies of representable ones, one copy for each element, along the maps between those elements.\nThis is like saying that every set is the union of its one-point subsets, but in a setting with arrows.\n\n### L2 human\n\nThe theorem is used to extend a functor $F \\colon \\mathcal{C} \\to \\mathcal{E}$ to all presheaves when $\\mathcal{E}$ has colimits: send $P$ to the colimit of $F \\circ \\pi$.\nThis extension is the left Kan extension of $F$ along $y$, and it is left adjoint to the functor $E \\mapsto \\Hom_{\\mathcal{E}}(F(-), E)$.\nGeometric realisation of simplicial sets is built this way, from the standard simplices.\nThe theorem also shows that $\\widehat{\\mathcal{C}}$ is the free cocompletion of $\\mathcal{C}$.\n",
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