Vladimir: if you take \$$f : \mathbb{N} \to \mathbb{N}\$$ to be multiplication by 10:

$$f(x) = 10 x$$

you can check that its right adjoint \$$g: \mathbb{N} \to \mathbb{N}\$$ is

$$g(x) = \lfloor \frac{x}{10} \rfloor$$

since for _all_ natural numbers \$$a\$$ and \$$b\$$

$$10 a \le b \textrm{ if and only if } a \le \lfloor \frac{b}{10} \rfloor .$$

(Check this!) This fits nicely with our intuition that the right adjoint is the "best approximation from below" to the nonexistent inverse of \$$f\$$.

**Puzzle.** What's the left adjoint of this function \$$f\$$?

But you can also do other things. For example, let \$$B\$$ be the set of natural numbers divisible by 10, and create a new function \$$F : \mathbb{N} \to B \$$ defined by

$$F(x) = 10 x$$

It looks like the same function, but it's different because it maps to a different set.

**Puzzle.** Does this function \$$F\$$ have a right adjoint? If so, what is it?

**Puzzle.** Does this function \$$F\$$ have a left adjoint? If so, what is it?