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 → 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, φ) where A is an object of D and φ: X → 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 → U(Y) is a morphism in C, then there exists a unique morphism g: A → Y such that the following diagram commutes:
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: