This may not be very useful, but since I had this thought while reading, I might as well post it.

I was wondering what you meant by *best* approximation, and I can see how this is a natural way of defining it, given that all we have is the partial (or pre)order. I was wondering though whether another type of *best* approximation might be about limiting the domain, rather than limiting the value the function takes. So for instance, the domain on which we have an inverse for the function \\(f: \mathbb{N} \to \mathbb{N}\\) with \\(f(n) = 2\cdot n \\) is \\(2 \cdot \mathbb{N}\\) (by which I mean the set of all even numbers). Thus, in that case I would get an approximation that is limited in its domain, but accurate, whereas the right and left adjoints are given on the full domain, but wrong in places.

After I thought a bit about it, I felt though that this is worse than the right and left adjoints, because the right and left adjoints together contain more information. I think (without having proved it) that the domain I was thinking of is the domain where the right and left adjoints agree in value -- so it has less information than the adjoints.

I was wondering what you meant by *best* approximation, and I can see how this is a natural way of defining it, given that all we have is the partial (or pre)order. I was wondering though whether another type of *best* approximation might be about limiting the domain, rather than limiting the value the function takes. So for instance, the domain on which we have an inverse for the function \\(f: \mathbb{N} \to \mathbb{N}\\) with \\(f(n) = 2\cdot n \\) is \\(2 \cdot \mathbb{N}\\) (by which I mean the set of all even numbers). Thus, in that case I would get an approximation that is limited in its domain, but accurate, whereas the right and left adjoints are given on the full domain, but wrong in places.

After I thought a bit about it, I felt though that this is worse than the right and left adjoints, because the right and left adjoints together contain more information. I think (without having proved it) that the domain I was thinking of is the domain where the right and left adjoints agree in value -- so it has less information than the adjoints.