Reading category theory book, Awodey states (during the introduction to Free monoids, p.18):
First, every monoid $N$ has an underlying set $|N|$, and every monoid homomorphism $f:N\to M$ has an underlying function $|f|:|N|\to |M|$. It is easy to see that this is a functor, called the "forgetful functor."
- Where is the "forgetful functor" the author referring to?
- Also, what does the notation with vertial lines around the function name $|f|$ means?
As I understand it, monoid homomorphism is simply a functor, but $|f|:|N|\to |M|$ is simply a function between sets, and not a functor... or it's a functor in Sets, but it's not forgetting anything.
Thank you for any clarification.