Keith wrote approximately:

> In the context of numbers, the adjoint is the multiplication of a natural number by \\(-1\\),

> \[ \begin{align} \mathbb{N} &\to \mathbb{Z} \\ n &\mapsto -n \end{align} \]

> its right adjoint is the absolute value,

> \[ \begin{align}\mathbb{Z} &\to \mathbb{N}\\ z &\mapsto |z| \end{align} \]

That's not true. Also, you seem to be making a level slip here. We are looking for adjoint functors

\[ L : \mathbf{Mon} \to \mathbf{Grp}, \qquad R: \mathbf{Grp} \to \mathbf{Mon} \]

but you seem to be trying to invent adjoint functors from a _particular_ monoid to a _particular_ group. A group is a kind of monoid, and a monoid is a kind of category, so it does make sense to talk about adjoint functors between monoids (e.g. groups). But that's not what we were talking about.

We can look into this issue anyway, just for fun:

A functor between monoids is just the same as a monoid homomorphism. This is an example of one:

> \[ \begin{align} \mathbb{N} &\to \mathbb{Z} \\ n &\mapsto -n \end{align} \]

but this is not:

> \[ \begin{align}\mathbb{Z} &\to \mathbb{N}\\ z &\mapsto |z| \end{align} \]

because \\(|z_1 + z_2| \ne |z_1| + |z_2| \\).

The only functor (i.e. monoid homomorphism) from \\(\mathbb{Z}\\) to \\(\mathbb{N}\\) is the one that sends every integer to zero. If you write down the definition of adjoint functor and apply it to this case, you'll see there are no adjoint functors between \\(\mathbb{N}\\) and \\(\mathbb{Z}\\).

> In the context of numbers, the adjoint is the multiplication of a natural number by \\(-1\\),

> \[ \begin{align} \mathbb{N} &\to \mathbb{Z} \\ n &\mapsto -n \end{align} \]

> its right adjoint is the absolute value,

> \[ \begin{align}\mathbb{Z} &\to \mathbb{N}\\ z &\mapsto |z| \end{align} \]

That's not true. Also, you seem to be making a level slip here. We are looking for adjoint functors

\[ L : \mathbf{Mon} \to \mathbf{Grp}, \qquad R: \mathbf{Grp} \to \mathbf{Mon} \]

but you seem to be trying to invent adjoint functors from a _particular_ monoid to a _particular_ group. A group is a kind of monoid, and a monoid is a kind of category, so it does make sense to talk about adjoint functors between monoids (e.g. groups). But that's not what we were talking about.

We can look into this issue anyway, just for fun:

A functor between monoids is just the same as a monoid homomorphism. This is an example of one:

> \[ \begin{align} \mathbb{N} &\to \mathbb{Z} \\ n &\mapsto -n \end{align} \]

but this is not:

> \[ \begin{align}\mathbb{Z} &\to \mathbb{N}\\ z &\mapsto |z| \end{align} \]

because \\(|z_1 + z_2| \ne |z_1| + |z_2| \\).

The only functor (i.e. monoid homomorphism) from \\(\mathbb{Z}\\) to \\(\mathbb{N}\\) is the one that sends every integer to zero. If you write down the definition of adjoint functor and apply it to this case, you'll see there are no adjoint functors between \\(\mathbb{N}\\) and \\(\mathbb{Z}\\).