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  "body": "\n## Summary\n\nA [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{C} \\to \\mathbf{Cat}$ assigns a category to each object of $\\mathcal{C}$.\nThe Grothendieck construction turns $F$ into a single [[category|A2DE66937D]] $\\int F$ with a projection functor to $\\mathcal{C}$.\nIt glues the categories $F(c)$ together along the functors $F(f)$.\nIt is the bridge between functors into $\\mathbf{Cat}$ and categories fibred over $\\mathcal{C}$.\nIt contains the category of elements of a set-valued functor as a special case.\n\n## Explanations\n\n### L1 human\n\nThink of $\\mathcal{C}$ as a base, and above each point $c$ of the base sits a whole category $F(c)$, called the fibre.\nThe Grothendieck construction puts all the fibres side by side and adds arrows that move from one fibre to another, following arrows of the base.\nAn object of $\\int F$ is a pair: a base object $c$ and an object $x$ of the fibre over $c$.\nFor a small example, if $F$ is constant with value a category $\\mathcal{D}$, then $\\int F$ is the [[product category|AFNE9C4NH6]] $\\mathcal{C} \\times \\mathcal{D}$.\nIf every fibre is a set with only identity arrows, the construction gives the category of elements of $F$.\n",
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