OK, I'll ask a very high level question, because I have shamefully been too busy to keep up with the technical detail for the last couple of weeks: how do we make this definition of limit yield the notion of a product (which I understand is a kind of limit). People who have taken at least part of a category theory course may have encountered the definition of a product via a universal property (https://en.wikipedia.org/wiki/Product_(category_theory)): how does this relate? I think once I understand this (and I'll need to work through the details) a lot of stuff will click.