Generator

Remarks

When $\mathcal{C}$ is locally small, $G$ is a generator exactly when the hom functor $\Hom(G, -) \colon \mathcal{C} \to \mathbf{Set}$ is faithful. The dual notion is a cogenerator (or coseparator): an object $C$ such that $f \neq g \colon X \to Y$ implies $k \circ f \neq k \circ g$ for some $k \colon Y \to C$.

Example: in the category of abelian groups, $\Z$ is a generator. If $f \neq g \colon A \to B$, pick $a \in A$ with $f(a) \neq g(a)$ and let $h \colon \Z \to A$ be $n \mapsto na$; then $(f \circ h)(1) = f(a) \neq g(a) = (g \circ h)(1)$.

Non-example: in the same category, the trivial group $0$ is not a generator. The identity and the zero map of $\Z/2\Z$ differ, but the only morphism $h \colon 0 \to \Z/2\Z$ is zero, and both composites with $h$ are zero.