How to take a limit of a diagram with more than one category? The German Wikipedia describes how one can define the quotient field of a ring over a universal property:


A quotient field $(\mathrm{Quot}(R), i)$ of a ring $R$ is a field $\mathrm{Quot}(R)$ together with an injective ring homomorphism $i:R\to\mathrm{Quot}(R)$ with the universal property that for each field $K$ and each injective ring homomorphism $f:R\to K$ exists a unique field homomorphism $g:\mathrm{Quot}(R)\to K$ such that $f=g\circ i$.

Intuitively, this makes sense: I imagine the quotient field to be some kind of "smallest field that includes the ring $R$".
However, I don't understand how to define this in a formal way: The formal definition of a universal property deals with limits: one defines a diagram as a functor $\mathcal{F}:D\to C$ from a finite category $D$ (with the diagram structure) to the category $C$ at hand, then one defines a cone to this functor, and the limit is a terminal object in the category of cones.
The limit we're interested in is not a ring, it is a field, so I would expect the diagram to be in the category of fields and field homomorphisms. But $R$ is not a field, so this cannot be true. But if the diagram is in the category of rings, then the limit is a ring instead of a field.
How does one treat limits of diagrams where the image of the diagram is in more than one category?
 A: The term "universal property" does not refer just to limits.  More generally, it refers to any structure that can be described as a terminal object in some category (probably not your original category).  For instance, the limit of a diagram is just a terminal object in the category of cones under the diagram.
In this case, the category in question is the category whose objects are fields $K$ together with a ring-homomorphism $R\to K$, and the morphisms are homomorphisms between the fields that make the diagram commute.  A quotient field of $R$ is then just an initial object in this category (or if you prefer, a terminal object in the opposite category).
As you observe, this definition is secretly talking about two different categories at once: the category of rings and the category of fields.  To make this more explicit, you can mention you are using the forgetful functor $F$ from the category of fields to the category of rings to identify a field $K$ as also giving a ring $F(K)$.  You can then say an object of your category is then a pair $(f,K)$ where $K$ is an object in the category of fields and $f$ is a morphism in the category of rings from $R$ to $F(K)$.  A morphism from $(f,K)$ to $(g,L)$ is then a morphism $h:K\to L$ in the category of fields such that $g=F(h)\circ f$.
A: Here the adequate notion is that of adjoint functors : there is the inclusion functor $U: Field\to Domain$ (the functor that forgets that a field is more than an integral domain).
The functor $Q: Domain\to Field$ that takes an integral ring to its field of fraction is the left adjoint to $U$ : you have $Hom(Q(R),F) = Hom(R,U(F))$ for any domain $R$ and any field $F$.
This is a case of a reflexive subcategory : this is when the inclusion funtor has a left adjoint.

You seem to think that the general framework for universal properties is the notion of limits, but actually it would be more the notion of representable functors (of which limits are a special case).
