TITLE: Why are there so many universal properties in math?
QUESTION [23 upvotes]: I don't really understand why there are so many universal properties in math or why they all need to be highlighted.
For example, I'm studying some Algebra right now.  I have found three universal properties that are all basically saying the same thing, although the details are different:

Universal property 1: If $R, S$ are rings and $\theta: R \to S$ is a ring map, then for each $s \in S$, there is a unique map $\hat{\theta_{s}} : R[x] \to S$ such that if $i: R \to R[x]$ is the inclusion map, we get $\theta = \hat{\theta_{s}} \circ i$.
Universal property 2: If $D$ is an integral domain and $F$ is a field with $\phi : D \to F$ a one-to-one ring map, then there is a unique map $\hat{\phi} : Q(D) \to F$ such that $\hat{\phi} \circ \pi = \phi$, where $\pi : D \to Q(D)$ sends $a$ to $\frac{a}{1}$ ($Q(D)$ the fractional field of $D$).

Universal property two was used to prove that in a field of characteristic $0$, the rationals are a subfield, and in a field of characteristic $p$ ($p$ prime), $\mathbb{Z}_{p}$ is a subfield.

Universal property 3: If $R, S$ are rings, $\phi: R \to S$ is a ring map, and $I$ is an ideal such that $I \subseteq \text{ker}(\phi)$, then there is a unique map $\overline{\phi} : R/I \to S$ such that $\phi = \overline{\phi} \circ i$ where $i: R \to R/I$ maps $a$ to $\overline{a}$.

It is really hard for me to keep track of all of these universal properties, especially when they are all usually referenced by the single name "universal property".  Is there a point to all of these universal properties?
Honestly, I don't even know if my question is clear, or how to ask a better question in this regard.

REPLY [3 votes]: I had the same question when I was learning algebra from Grillet‘s Algebra book (GTM 242). At that time, I did understand how I could define concepts (i.e. tensor products, free products and free groups, etc.) with universal property (UP), how maps factor through one another, and how the object defined is the “most general of all”. However, I failed to see how UP is important in other contexts (quotients in topology/groups in particular) even though Grillet stated a lot of theorems in UP, as I felt that it is at most a fancy description with commutative diagrams. And yes, I resonate with you when you said "it is really hard for me to keep track of all of these UPs".
However, now that I've learned more, I found that UP is a very useful tool when you are given a map $f: X \rightarrow Y$ and you are trying to find a corresponding map from $\tilde{f}: \tilde{X} \rightarrow Y$, where $\tilde{X}$ is some object you've constructed from $X$ with the map $\phi : X \rightarrow \tilde{X}$. As long as $\tilde{X}$ and $f$ is well behaved (i.e. satisfies the hypothesis for UP), you can get a unique $\tilde{f}$ with similar good properties that $f$ possesses. Even better, you get to factor through the map from $X \rightarrow \tilde{X}$.
This idea might sound cliche, since the proofs of UP usually involve constructions of $\tilde{f}$ induced by $f$ in the most obvious way. But in many more advanced subjects, we are defining the $\tilde{X}$'s (some of which seemed very messy) from intuitive spaces like $X$, while we only know about maps from $X \rightarrow Y$. With universal property, you don't need to explicitly state what $\tilde{f}$ is and simply acknowledge its existence and uniqueness, and this is extremely helpful if you are juggling with both spaces and trying to construct such maps a couple of times in one proof. I realized this when I was learing algebraic topology and proving homotopy equivalence of a space with its quotient space (under certain conditions), and it was then when I suddenly realized the power of the universal property in quotient topology.
A final note on "hard to keep track of all UP's": I don't think you need to actually remember the statements of all UP's, because as long as you know how $\tilde{f}$ is constructed, you could easily recall the UP when you need it. It's okay to find it useless when you encounter statements like this for the first time, but keep in mind that UP will be of much greater significance as you learn more mathematics.