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  "body": "\n## Summary\n\nAn end is a universal way of extracting from a [[functor|28NQ0KRHD3]] $S \\colon \\mathcal{C}^{\\mathrm{op}} \\times \\mathcal{C} \\to \\mathcal{D}$ an object that is compatible with all the diagonal values $S(c, c)$.\nA coend is the dual notion.\nEnds are special [[limits|CHS5AFJGWE]] and coends are special [[colimits|BJS1CGTKQ5]].\nThe set of [[natural transformations|KYCPCC57FT]] between two functors is an end, and many constructions such as tensor products and Kan extensions are coends or ends.\n\n## Explanations\n\n### L1 human\n\nSome functors take two inputs from the same category, one in a backward and one in a forward way, like $\\Hom(c, c')$.\nAn end picks out the families of diagonal elements, one for each object $c$, that are consistent along all arrows.\nFor example, a natural transformation between two functors is a family of maps, one per object, consistent with all arrows.\nSo the natural transformations form the end of the functor $(c, c') \\mapsto \\Hom(F c, G c')$.\nA coend glues the diagonal values together instead, and plays the role of a tensor product.\n",
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  "name": "End and coend",
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