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  "body": "\n## Summary\n\nDay convolution is a tensor product on the category of set-valued functors on a small [[monoidal category|M3MSQ3NKAE]].\nIt is built from the tensor product of the source category.\nIt makes the [[functor category|1X0WKFQX8K]] into a closed monoidal category.\nThe Yoneda embedding becomes a strong monoidal functor.\nSo Day convolution is the universal way to extend a monoidal structure from a category to its category of set-valued functors.\nIt generalizes the convolution of functions on a group and the Cauchy product of power series.\n\n## Explanations\n\n### L1 human\n\nThink of a functor $F$ from a monoidal category to sets as a family of sets $F(a)$, one for each object $a$.\nTo combine two such families at an object $c$, one collects all pairs of elements $x \\in F(a)$, $y \\in G(b)$ together with a way to map $a \\otimes b$ to $c$.\nFor example, if the objects are the natural numbers with $+$ and only identity arrows, a functor is a graded set.\nThe Day convolution of two graded sets has in degree $n$ the pairs of elements whose degrees add up to $n$, exactly like multiplying power series.\n",
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