Fredrick wrote:

> ...what do we do when \$$x \lt 0 \$$?

We have a formula for the right adjoint if it exists: we saw it near the end of [Lecture 6](https://forum.azimuthproject.org/discussion/1901/lecture-6-chapter-1-computing-adjoints/p1). So, we can use this to figure out what \$$R(x)\$$ must be if the right adjoint exists... and if the formula gives an undefined result, we know the right adjoint cannot exist.

Another approach is to use Proposition 1.81 in _[Seven Sketches](http://math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf)_. Applied to our example, this implies that if \$$R : \mathbb{R} \to \mathbb{N}\$$ is a right adjoint to \$$I : \mathbb{N} \to \mathbb{R} \$$, we must have

$$I(R(x)) \le x$$

for all \$$x \in \mathbb{R}\$$. See what this means?