# 📝Category Theory: Epimorphism

\begin{tikzcd} X \arrow[r, "f"] & Y \arrow[r, "g_1", shift left] \arrow[r, "g_2", shift right, swap] & Z \end{tikzcd}

$f : X \to Y$ is an epimorphism iff $\forall(g_1, g_2 : Y \to Z)\; g_1 \circ f = g_2 \circ f \implies g_1 = g_2$

(The rule to remember is that you can cancel $f$ from the right side of equation.)

Epimorphism is defined for all categories (but might be absent), and is a generalization of surjective functions from Set Theory (Set theory: functions).

See also: