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  "body": "\n## Summary\n\nA profunctor, also called a distributor or bimodule, from a category $\\mathcal{C}$ to a category $\\mathcal{D}$ is a [[functor|28NQ0KRHD3]] $\\mathcal{D}^{\\mathrm{op}} \\times \\mathcal{C} \\to \\mathbf{Set}$.\nIt generalizes functors: every functor gives a profunctor.\nIt also generalizes relations: a relation between sets is a profunctor between discrete categories with values in truth values.\nProfunctors compose by a [[coend|W9QTFV0Z8D]], and categories, profunctors and natural transformations form a bicategory.\nThey are the categorical analogue of bimodules between rings.\n\n## Explanations\n\n### L1 human\n\nA function from $X$ to $Y$ assigns to each $x$ one $y$, while a relation only says which pairs $(x, y)$ are related.\nA profunctor does the same for categories, and it says how $x$ and $y$ are related by a set of \"witnesses\" $P(d, c)$.\nThe witnesses can be moved along morphisms in both categories, in opposite directions.\nFor example, the hom-sets $\\Hom(d, c)$ of a category form a profunctor from the category to itself: a morphism is a witness that $d$ maps to $c$.\n",
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