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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be [[categories|A2DE66937D]], and let $\\mathcal{D}^{\\mathrm{op}}$ be the [[opposite category|DAMK8XE1FR]] of $\\mathcal{D}$.\nA *profunctor* $P$ from $\\mathcal{C}$ to $\\mathcal{D}$, written $P \\colon \\mathcal{C} \\nrightarrow \\mathcal{D}$, is a [[functor|28NQ0KRHD3]]\n$$P \\colon \\mathcal{D}^{\\mathrm{op}} \\times \\mathcal{C} \\to \\mathbf{Set}$$\non the [[product category|AFNE9C4NH6]].\nA morphism of profunctors is a natural transformation between such functors.\n\n## Remarks\n\nConventions differ: some authors call the same functor a profunctor from $\\mathcal{D}$ to $\\mathcal{C}$.\nFor small $\\mathcal{D}$ it is the same as a functor $\\mathcal{C} \\to [\\mathcal{D}^{\\mathrm{op}}, \\mathbf{Set}]$, from $\\mathcal{C}$ to presheaves on $\\mathcal{D}$.\nExample: the [[hom functor|VJXS8AK67E]] $\\Hom_{\\mathcal{C}} \\colon \\mathcal{C}^{\\mathrm{op}} \\times \\mathcal{C} \\to \\mathbf{Set}$ is a profunctor from $\\mathcal{C}$ to $\\mathcal{C}$, the identity profunctor.\nExample: a functor $F \\colon \\mathcal{C} \\to \\mathcal{D}$ gives the profunctor $(d, c) \\mapsto \\Hom_{\\mathcal{D}}(d, F(c))$ from $\\mathcal{C}$ to $\\mathcal{D}$.\nNon-example: for the same $F$, the functor $(d, c) \\mapsto \\Hom_{\\mathcal{D}}(F(c), d)$ is covariant in $d$ and contravariant in $c$.\nSo it is not a profunctor from $\\mathcal{C}$ to $\\mathcal{D}$; it is a profunctor from $\\mathcal{D}$ to $\\mathcal{C}$.\nFor $P \\colon \\mathcal{C} \\nrightarrow \\mathcal{D}$ and $Q \\colon \\mathcal{D} \\nrightarrow \\mathcal{E}$ with $\\mathcal{D}$ small, the composite is given by the [[coend|W9QTFV0Z8D]] $(Q \\circ P)(e, c) = \\int^{d \\in \\mathcal{D}} Q(e, d) \\times P(d, c)$.\nComposition is associative and unital only up to natural isomorphism.\n",
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