$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 10.3.4.9. Let $X$ and $Y$ be sets, and let $f: X \rightarrow Y$ be a function. The following conditions are equivalent:
- $(1)$
The function $f$ is a universal quotient morphism in the category of sets.
- $(2)$
The function $f$ is a quotient morphism in the category of sets.
- $(3)$
The function $f$ is surjective.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Remark 10.3.4.2 and the equivalence $(2) \Leftrightarrow (3)$ follows from Example 10.3.2.8. Since the collection of surjections is closed under pullbacks, Corollary 10.3.4.7 guarantees that $(3) \Rightarrow (1)$.
$\square$