Example 10.3.3.4 (Images of Sets). Let $f: X \rightarrow Y$ be a function between sets, and set $Y_0 = \{ y \in Y: f^{-1} \{ y\} \neq \emptyset \} $. Then $f$ determines a surjection from $X$ to $Y_0$, which is a quotient morphism in the category of sets (Example 10.3.2.8). It follows that the commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{f} \ar [rr]^{f} & & Y } \]
exhibits $Y_0$ as an image of $f$.