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?

$$ f(x) = 10 x $$

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?