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  "body": "\n## Definition\n\nLet $(\\mathcal{C}, \\otimes, I)$ be a small [[monoidal category|M3MSQ3NKAE]], and let $[\\mathcal{C}, \\mathbf{Set}]$ be the [[functor category|1X0WKFQX8K]] of functors $\\mathcal{C} \\to \\mathbf{Set}$.\nThe *Day convolution* of $F, G \\in [\\mathcal{C}, \\mathbf{Set}]$ is the functor $F \\ast G \\colon \\mathcal{C} \\to \\mathbf{Set}$ given by the [[coend|W9QTFV0Z8D]]\n$$(F \\ast G)(c) = \\int^{a, b \\in \\mathcal{C}} \\Hom_{\\mathcal{C}}(a \\otimes b, c) \\times F(a) \\times G(b).$$\nWith unit object the [[hom functor|VJXS8AK67E]] $\\Hom_{\\mathcal{C}}(I, -)$, Day convolution makes $[\\mathcal{C}, \\mathbf{Set}]$ a monoidal category.\n\n## Remarks\n\nEquivalently, $F \\ast G$ is the left [[Kan extension|RN9D96ESWN]] of the functor $(a, b) \\mapsto F(a) \\times G(b)$ on $\\mathcal{C} \\times \\mathcal{C}$ along $\\otimes \\colon \\mathcal{C} \\times \\mathcal{C} \\to \\mathcal{C}$.\nNatural transformations $F \\ast G \\to H$ correspond to families of maps $F(a) \\times G(b) \\to H(a \\otimes b)$ natural in $a$ and $b$.\nExample: let $\\mathcal{C}$ be the discrete category on $\\N$ with $\\otimes = +$ and $I = 0$, so a functor $F$ is a sequence of sets $F(0), F(1), \\ldots$.\nSince $\\Hom(a + b, n)$ is a point when $a + b = n$ and empty otherwise, $(F \\ast G)(n) = \\bigsqcup_{a + b = n} F(a) \\times G(b)$, the Cauchy product.\nNon-example: the pointwise product $(F \\times G)(c) = F(c) \\times G(c)$ is also a monoidal structure, but it is not the Day convolution.\nIn the example above it pairs elements of equal degree, and its unit is the constant functor with value a point, not $\\Hom(0, -)$.\nThe monoidal category $[\\mathcal{C}, \\mathbf{Set}]$ with Day convolution is [[closed|EKJBDD07TY]], and $a \\mapsto \\Hom_{\\mathcal{C}}(a, -)$ is a strong monoidal functor $\\mathcal{C}^{\\mathrm{op}} \\to [\\mathcal{C}, \\mathbf{Set}]$.\nThe same construction works for functors into a cocomplete closed symmetric monoidal category $\\mathcal{V}$ in place of $\\mathbf{Set}$.\n",
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