I want to understand how the following observation applies to free algebraic systems. In Thm. II.7.1 of Mac Lane's Categories for the Working Mathematician (p.49 of ed. 2), the functor $P:G \rightarrow UC$ is universal when $U$ the forgetful functor. The theorem:
Check me here: for functor $U$ and object $G$ (are we treating $G$ as an object in the category of categories?), the pair $\langle C, P\rangle$ is a universal arrow from $G$ to $U$ because for every other pair $\langle B,D \rangle$ where $D: G \rightarrow UB$, $D$ can be factored as $(UD') \circ P = D$. Correct?
On p.56 Mac Lane states that this same idea can be applied to free algebraic systems which I don't understand.
Monoid case: should I be thinking of $C$ and $B$ as free monoids (understood as categories with single objects) with a functor $D'$ between them? Is $G$ for this case then the small graph with a single object and arrows the self loops corresponding to the generators of $C$? A careful explanation of this case would be helpful for me.
Is the takeaway that free monoid $C$ and functor $U$ are somehow special because of the factoring condition? (The concept of universality is a new one for me.)
Thanks in advance!
To clarify the connection between free algebraic structures (like monoids) and the theorem you quoted, let me write out the analogous theorem about monoids. I'll write $U$ for the "underlying set" functor from monoids to groups.
"Let $G$ be a set. There is a monoid $C=C_G$ and a morphism $P:G\to UC$ of sets from $G$ to the underlying set $UC$ of $C$ with the following property. Given any monoid $B$ and any morphism $D:G\to UB$ of sets, there is a unique morphism of monoids $D':C\to B$ with $(UD')\circ P=D$ as in the commutative diagram [same diagram as in Mac Lane's theorem]."
This is almost word-for-word the theorem you quoted. "Set" and "monoid" have replaced "graph" and "category". Remember that morphisms of sets, monoids, and categories are functions, homomorphisms, and functors, respectively.
You should check that the $C$ is the monoid version of the theorem can be constructed as the set of all finite words on the alphabet $G$ with the operation of concatenation (i.e., the usual construction of a free monoid on $G$). The function $P$ sends each letter in the alphabet $G$ to the corresponding one-letter word in $C$.