Here's the definition of a universal property in Wikipedia:
(where $U:D\to C$ is a functor and $X$ is an object in $C$)
A terminal morphism from $U$ to $X$ is a final object in the category $(U\downarrow X)$ of morphisms from $U$ to $X$, i.e. consists of a pair $(A,\Phi)$ where $A$ is an object of $D$ and $\Phi: U(A) \to X$ is a morphism in $C$, such that the following terminal property is satisfied:
- Whenever $Y$ is an object of $D$ and $f: U(Y) \to X$ is a morphism in $C$, then there exists a unique morphism $g: Y \to A$ such that the following diagram commutes:
So I'm trying to "unpack" this definition and figure out what each of the things here "means". E.g. what does it become in the case of a limit, or something.
- A limit is an example of a terminal morphism, right? And a colimit an initial morphism?
- Does $U$ usually represent a diagram? In the case of a limit, does it represent the diagram we want to take the limit of?
- Whatever is $X$? I honestly have no clue here. What's the analog in the case of a limit?
- What does a morphism from $U\to X$ even mean? What does it mean in the case of a limit? I've seen morphisms from a diagram to an object in colimits.
- In the case of a limit, is $(U\downarrow X)$ the category of cones? But how can each cone be a morphism from $U$ to something (I thought it was a morphism from something to $U$)?
- $A$ (or $U(A)$) corresponds to the actual thing we construct, like the source of a limit or the target of a colimit? But what is $\Phi$? In the construction of a limit, there's a morphism from the limit to the diagram, this seems wrong.
My guess is that $X$ represents some sort of "subsetting" of the candidates for the object so you don't have to quantify over everything like you do with cones and the limit. Is that right?
Edit: So as a short summary -- it turns out $X$ represents (in the case of limits and colimits) the diagram we're trying to take the limit of, while $A$ represents the actual limit object (with its morphism $\Phi$). $U$ is the diagonal functor, because the limit is constructed here as an object in the category of diagrams of shape at most that of $X$.