Universal Property for Field of Quotients

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain.

Let $\left({F, \oplus, \cdot}\right)$ be a quotient field of $D$.

Then $F$ satisfies the following universal property:

There exists a homomorphism $\iota : D \to F$ such that for every field $\tilde F$ and for every homomorphism $\phi : D \to \tilde F$, there exists a unique homomorphism $\psi : F \to \tilde F$ satisfying $\psi \iota = \phi$. That is, the following diagram commutes:


 * FieldFracComDiag.jpg

Namely we may take $\psi : a/b \mapsto \phi(a)\phi(b)^{-1}$.