# What does Universal mapping property for a free monoid mean?

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)

1. What does the author want me to learn here?
2. Where is the definition of the "Free monoid" - it seems to me like he is referring to an early definition in "by definition".
3. 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.

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1. The author is introducing a definition of a mathematical object, not by constructing it explicitly, but by describing an important property that it satisfies. It is a non-obvious fact that this important property characterizes the object in question "up to unique isomorphism" (which you can read as meaning "uniquely" for the time being).

2. The universal mapping property is the definition. It is a non-obvious fact that this is a meaningful way to define something.

3. "The" is shorthand for "unique up to unique isomorphism." I wouldn't worry about this for the time being.

Universal properties can be thought of as a vast generalization of the notion of "largest" or "smallest." In many cases they can be thought of as the "laziest" way to do something. In this case, the free monoid can be thought of as the "laziest" way to turn a set into a monoid. This will be made clearer by a more explicit description of the free monoid (I am assuming that Awodey gives such a description) as the set of words on the elements of $A$.

You might find it helpful to supplement Awodey by reading Lawvere and Schanuel's Conceptual Mathematics.

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Amazing book recommendation, thank you :-) –  drozzy Oct 30 '12 at 7:13
I think I kind of understand what you mean by laziest. Eugenia Cheng explains a "free monoid functor" in her Category Theory 3 (p. 4, pdf). So can there be two different "free monoids" for a given underlying set? –  drozzy Oct 30 '12 at 7:45
Unfortunately Conceptual Mathematics does not seem to have anything on free monoids... –  drozzy Oct 30 '12 at 7:56
@drozzy: yes, but there is a unique isomorphism between them which is compatible with the inclusion maps (the one above is denoted by $i$). I don't remember if Conceptual Mathematics has anything about free monoids in particular, but it has some material about universal properties in general, and I think you will find reading it indirectly if not directly helpful. –  Qiaochu Yuan Oct 30 '12 at 7:59
Sorry to keep bugging you, but could you ever construct a monoid that does not satisfy a Universal Mapping Property? –  drozzy Nov 13 '12 at 4:05