Why is the underlying homomorphism function a "forgetful functor"? 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."



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*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.

The " forgetful functor" is the map 
$$N \mapsto |N| \text{ the underlying set of N }$$ 
$$f \mapsto |f| \text{ the morphism viewed as a map of sets }$$ 
Its a functor from the category of monoids to the category sets. And it is forgetting about the properties of being a monoid and just stripping down to the underlying set. 
