Uniqueness of homomorphism

Question

$\DeclareMathOperator{\Frac}{Frac}$How does one prove the uniqueness of a homomorphism? My case is about the Universal Property of Fraction Fields. Let $R$ be an integral domain, $\iota: R\to \Frac(R)$ is the inclusion homomorphism taking $r$ to $r/1$. I have the homomorphism $\psi: \Frac(R)\to F$, where $\Frac(R)$ is an injective ring homomorphism defined as $\psi(a/b) = \phi(a)\phi^{-1}(b)$, where $\phi: R\to F$ is another injective (not explicitly defined) homomorphism. ($R$ is a ring). It is known that $\psi$ is injective and that it must satisfy $\psi\circ\iota=\phi$ (and it does). But how does one prove that $\psi$ is actually unique?

Answer 1

Suppose $\chi\colon\operatorname{Frac}(R)\to F$ is another ring homomorphism such that $\chi\circ\iota=\phi$. Then for any $r\in R$, $$ \chi(r/1)=\chi\iota(r)=\phi(r)=\psi\iota(r)=\psi(r/1). $$

Also, for any $r\neq 0$ in $R$, $$ \chi(1/r)=\chi((r/1)^{-1})=\chi(r/1)^{-1}=\psi(r/1)^{-1}=\psi(1/r). $$

Then for $r\in R$, and $s\in R$ nonzero, $$ \chi(r/s)=\chi\left(\frac{r}{1}\cdot\frac{1}{s}\right)=\chi(r/1)\chi(1/s)=\psi(r/1)\psi(1/s)=\psi(r/s). $$