I am going trough the "Category Theory" book by Steve Awodey.
In the "1.7 Free categories" chapter the author introduces the following algebraic definition of free monoid:
A monoid $M$ is freely generated by a subset $A$ of $M$, if the following conditions hold:
(no-junk) Every element $m \in M$ can be written as a product of elements of $A$ $$m = a_1 \cdot_M a_2 \cdot_M...\cdot_M a_n, \space a_i\in A$$
(no-noise) No "nontrivial" relations hold in $M$, that is, if $a_1...a_j = a_1^`...a_k^`$, then this is required by the axioms for monoids
Then the autor introduces the notion of $Universal \space Mapping \space Property \space (UMP)$ as a way to encode the conditions above in terms of category theory:
Let $M(A)$ be a monoid on a set $A$. There is a function $i: A \rightarrow|M(A)|$ ($|M(A)|$ - is the underlying set of the $M$ monoid), and given any monoid $N$ and any funciton $f: A \rightarrow |N|$ ($|N|$ - is the underlying set of the $N$ monoid), there is a $unique$ monoid homomorphism $\bar f: M(A) \rightarrow N$ such that $|\bar f| \circ i = f$ where $|\bar f| : |M(A)| \rightarrow |N|$
The author then says that
- the existence part of the $UMP$ captures the vague notion of "no-noise" (because any equation that holds between algebraic combinations of the generators must also hold anywhere they can be mapped to, then thus everywhere)
- the uniqueness part makes precise the "no-junk" idea (because any extra elements non combined from the generators would be free to be mapped to different values)
None of the conclusion above seem to be clear to me, could anyone please explain it?