What is a universal property? Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia.

To make the following summary more readable I do equate "universal" with "initial" and omit the tedious details concerning duality.


  Suppose that $U: D \to C$ is a functor from a category $D$ to a category $C$,
  and let $X$ be an object of $C$.
  
  A universal morphism from $X$ to $U$ [...] consists of a pair $(A,\varphi)$
  where $A$ is an object of $D$ and $\varphi: X \to U(A)$ is a morphism in $C$, such
  that the following universal property is satisfied:
  
  Whenever $Y$ is an object of $D$ and $f: X \to U(Y)$ is a morphism in $C$, then
  there exists a unique morphism $g: A \to Y$ such that the following
  diagram commutes:
  
  $\hspace{5cm}$ 


What kind of definition is this? Instead of "such that the following universal property is satisfied" one can equivalently say "such that the following property is satisfied". So how can this be a definition of "universal property"?

Unfortunately, not even Awodey in his Category Theory gives a concise definition of "universal property".


  Where do I find a really concise definition of "universal property"?




EDIT: I wonder why the attitude "you only have to understand the concrete examples, and the abstract notion will pop out by itself" seems to be accepted in this context. This reminds me of Augustine of Hippo:  


  What, then, is time a universal property? If no one ask of me, I know; if I wish to
  explain to him who asks, I know not.
I agree with you that this is not about “concrete examples.” More about language. I apologize if my story is elementary, but there is really nothing complex.

Maybe you do not realize that “$A$ has universal property” is the same as “$A$ is a universal object” is the same as “an object $A$ is universal.” These are different names of the same term. So the definition of a universal object also defines universal property.

Consider the definition of an initial object. “…an initial object is an object… such that…” “Initial” is a property of objects, thus this definition defines a property. Properties are named not only by adjectives (e.g. “transitive”, “injective”), but also by nouns (e.g. “equivalence”, “injection”; “a function $f$ is an injection” is the same as “a function $f$ is injective”). In contrast, the definition of average, i.e. $(x, y)\mapsto \frac{x+y}{2}$, defines not a property.

Consider the shorter definition in Wikipedia which you did not cite:


  An initial morphism from $X$ to $U$ is an initial object in the category
  $X \downarrow U$.


This definition defines a property because it uses the definition of an initial object. The longer definition in Wikipedia which you cited is the shorter definition with the terms “initial object” and “comma category” unfolded.

“The universal property of the quotient group” is not a definition, it is a theorem which says that the quotient group $G/N$ is an initial object in a category defined as:


object: $(X, f)$ where $X$ is a group and $f:G\to X$ and $N\subseteq ker(f)$;
morphism of type $(X_0, f_0)\to (X_1, f_1)$: $g:X_0\to X_1$ such that $g\circ f_0 = f_1$.


I have essentially seconded lhf's answer, but he/she did not construct the category. I just can not find explicit construction of this category in textbooks.

Wikipedia's definition of the universal property does not include the universal property of the quotient group as a particular case. The problem is that in Wikipedia's definition $f$ is a morphism, but in the case of groups $f$ is a homomorphism such that $N\subseteq ker(f)$. IMHO Wikipedia's definition is not general enough.

P.&nbsp;S. I prefer “initial” and “terminal” over “universal”. A universal object is an initial object or a terminal object depending on context. Therefore, any text involving “universal” forces a reader to guess a precise meaning.
Given a diagram or graph a universal property says for the pair (U f) U is a object and f is a unique arrow, a U-cone is terminal or intial in f, if for any other cone over the U-cone the latter must factor through the former in particular through the unique arrow f.