In the nLab, Day convolution is introduced as a generalisation of convolution of complex-valued functions, but I'm wondering how exactly to understand this. I can (just about) parse the definitions, but have absolutely no intuition or geometric insight at all. Here is my thinking so far (excuse all of the quotation marks) :

A way of thinking about simple examples of presheaves is to imagine an association: to each open set of a topological space associate a set of continuous functions on that open set (really we could obtain a group or ring structure on this set of functions, but let's ignore that for a moment and just look at set-valued presheaves).

Then a presheaf is a 'function' that maps an open set to a set of functions. So a convolution of presheaves is obtained by 'blurring' these 'functions' together.

But what this actually means is rather beyond me. The issue (for me) is twofold:

- What does Day convolution look like in this simple case where we take our starting category to be $\mathsf{Op}(T)$, which is the category of open sets and inclusion maps of some topological space $T$ (which is usually the motivating example for presheaves).
**In fact, can we even look at this example?**As far as I can see, $\mathsf{Op}(T)$ doesn't admit a monoidal structure; **What does Day convolution look like generally?**Given some convolution of presheaves, are there any simple examples (if 1. doesn't work) that give a good intuition, where the Day convolution has a reasonably succinct description?

Edit:

- How does Day convolution fit in with 'regular' convolution.
That is,
**can we recover the usual convolution from Day convolution?**

I hope this question is rigorous enough, and if not then I'll try to edit it to be more so.