# 📝Category Theory: Monomorphism

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

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

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

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

See also: