The Stacks project

Definition 4.13.1. Let $\mathcal{C}$ be a category and let $f : X \to Y$ be a morphism of $\mathcal{C}$.

1. We say that $f$ is a monomorphism if for every object $W$ and every pair of morphisms $a, b : W \to X$ such that $f \circ a = f \circ b$ we have $a = b$.

2. We say that $f$ is an epimorphism if for every object $W$ and every pair of morphisms $a, b : Y \to W$ such that $a \circ f = b \circ f$ we have $a = b$.