Functions in the definition of Universal Mapping Property of a free monoid In Awodey's Category Theory (http://www.amazon.com/dp/0199587361/?tag=stackoverfl08-20) p.19, 
what is $|\bar{f}|$? What is its relation to $\bar{f}$, which is in the existence statement? Is it induced by $\bar{f}$?  
Moreover, what is $|M(A)|$? Is it the cardinal number of $M(A)$?  

It is the underyling set function obtained from $\overline f$. 
In detail, we are given a set $A$, we obtain the free monoid $M(A)$ in the usual way: The $x\in M(A)$ are words of finite length; the multiplication is cancatenation and the identity is the empty word denoted by $e$. 
On the other hand, $M(A)$ is also just a set so we can forget about its structure as a monoid and just look at its set properties. When we take this perspective, we call this set $\vert M(A)\vert$.We have the evident set map $$i:A\rightarrow \vert M(A)\vert $$ which just sends $a\in A$ to the word (of length 1) $a\in M(A)$ but $\textit forgetting$ the monoid multiplication.
Similarly, if we have a monoid homomorphism $$f:M\rightarrow N$$ we can forget for the moment that it is a homomorphism and just look at it as a set map. When we do this, we write it as $$\vert f\vert :\vert M\vert \rightarrow \vert N\vert $$
There is a canonical way to describe all this: take a set $A$ and form the free monoid $M(A)$.
Now suppose we have a $\textit set $ $A$ and an $$f:A\rightarrow \vert N\vert $$ Then Awodey proves that we can find a $\textit monoid$ $\text homomorphism$, $$\overline f:M(A)\rightarrow N$$ such that $$i \circ \vert \overline f\vert=f$$ where $\vert \overline f\vert$ is just $\overline f$ itself, but forgetting that $\overline f$ is a monoid homomorphism. 
In other words, $f$ which is just a set map, from a set $A$ to the set $\vert N\vert $ lifts to a unique monoid homomorphism from the free monoid $M(A)$ to the monoid $N$. 
Hope this helps.
