Bartosz Milewski on Nostr: It's a very deep analogy. Just like an integral, a coend is a sum: It's a coproduct ...

It's a very deep analogy. Just like an integral, a coend is a sum: It's a coproduct modulo some identifications, taking account of the arrows in the category. The x occurs in positive and negative positions (covariant/contravariant) just like in a convolution or Einstein summation. A (co-)presheaf F is like a vector indexed by x, in which case the coend is like a trace, and C(x, a) is like a Kronecker delta. In fact it becomes a delta if the category is discrete. Should I go on... ?