I am struggling to understand the "meaning" behind the Universal Mapping Property, as defined by Awodey (p.19):
The free monoid $M(A)$ on a set $A$ is by definition "the" monoid with the following so called universal mapping property or UMP!
Universal Mapping Property of $M(A)$
There is a function $i:A\to |M(A)|$, and given any monoid $N$ and any function $\bar{f}\circ i =f$, all as indicated in the following diagram: (... in Mon and in Sets diagrams follow)
- What does the author want me to learn here?
- Where is the definition of the "Free monoid" - it seems to me like he is referring to an early definition in "by definition".
- What does the author mean by the word "the" in "the moniod" - simply that it is unique?
I am altogether confused, and I would appreciate any an explanation or a different point of view on this matter. Thank you!
P.S.: Math level - novice.