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.
Best Answer
A universal property of some object $A$ tells you something about the functor $\hom(A,-)$ (or $\hom(-,A)$, but this is just dual). For example, $\hom(R[x],S) \cong |S| \times \hom(R,S)$ is the universal property of the polynomial ring (where $|S|$ denotes the underlying set of $S$). Conversely, we may consider the functor which takes a commutative ring $S$ to $|S| \times \hom(R,S)$ and say that it is a representable functor, represented by $R[x]$. This can be also interpreted as the statement that $R[x]$ is the free commutative $R$-algebra on one generator, see free object for categorical generalizations. Roughly, representing a functor means to give a universal example of, or to classify, the things which the functor describes. This happens all the time in mathematics. Conversely, whenever you have an object $A$, it is interesting to ask what it classifies, i.e. to look at $\hom(A,-)$ and give a more concise description of it. The Yoneda Lemma tells you that all information of $A$ is already encoded in $\hom(A,-)$.
Also, one of the main insights of category theory is that it is very useful to work with morphisms instead of elements. For example, what the quotient ring $R/I$ does for us is not really that we can compute with cosets, but rather that it is the universal solution to the problem to enlarge $R$ somehow to kill (the elements of) $I$. In other words, $\hom(R/I,S) \cong \{f \in \hom(R,S) : f|_I = 0\}$. This makes things like $(R/I)/(J/I) = R/J$ for $I \subseteq J \subseteq R$ really trivial: On the left side, we first kill $I$ and then $J$, which is the same as to kill $J$ directly, which happens on the right hand side. No element calculations are necessary. (On math.stackexchange, I have posted lots of examples for this kind of reasoning.) Quotient rings, quotient vector spaces, quotient spaces etc. are all special cases of colimits.
The universal property of the field of fractions states that $\hom(Q(D),F) \cong \hom(D,F)$, where on the right hand side we mean injective homomorphisms. This says that $Q(-)$ is left adjoint to the forgetful functor from fields to integral domains (in each case with injective homomorphisms as morphisms). This is a special case of localizations. Adjunctions are ubiquitous in modern mathematics. They allow us to "approximate" objects of a category by objects of another category.
So far I have only mentioned some patterns of universal properties, but not answered the actual "philosophical" question "Why are there so many universal properties in math?" in the title. Well first of all, they are useful, as explained above. Also notice that many objects of interest turn out to be quotients of universal objects. For example, every finitely generated $k$-algebra is a quotient of a polynomial algebra $k[x_1,\dotsc,x_n]$. Thus, if we understand this polynomial algebra and its properties, we may gain some information about all finitely generated $k$-algebras. A specific example of this type is Hilbert's Basis Theorem, which implies that finitely generated algebras over fields are noetherian. Perhaps one can say: Universal objects are there because we have invented them in order to study all objects.