Day convolution

In mathematics, specifically in category theory, Day convolution is an operation on functors that can be seen as a categorified version of function convolution. It was first introduced by Brian Day in 1970 in the general context of enriched functor categories. Day convolution acts as a tensor product for a monoidal category structure on the category of functors formula_1 over some monoidal category formula_2.
Definition.
Let formula_3 be a monoidal category enriched over a symmetric monoidal closed category formula_4. Given two functors formula_5, we define their Day convolution as the following coend.
If formula_7 is symmetric, then formula_8 is also symmetric. We can show this defines an associative monoidal product.