> So then, why is John using infs and sups when defining the unique formulas in lecture 6?

For complete lattices such as power set algebras and \\(\mathbb{R}\\), those characterize adjoints.

But as I try to show in **Puzzle MD 4** (where I consider the natural numbers ordered by the *evenly divides* relation), you can have left and right adjoints even when you can't take infima and suprema.

However, you can cheat out infima and suprema even if they don't exist by using Dedekind-Macneil completions. I did this over in the [Categories for the Working Hacker](https://forum.azimuthproject.org/discussion/comment/16649/#Comment_16649) discussion. I can write a formal proof regarding them and Galois connections if you like.