Implication Details

Assumptions: coproducts, generating set, zero morphisms

Conclusions: generator

Reason: If $S$ is a generating set, we claim that $U := \coprod_{G \in S} G$ is a generator. Let $f,g : A \rightrightarrows B$ be two morphisms with $f h = g h$ for all $h : U \to A$ . If $G \in S$ , any morphism $G \to A$ extends to $U$ by using zero morphisms outside of $G$ . Thus, $fh = gh$ holds for all $h : G \to A$ and $G \in S$ . Since $S$ is a generating set, this implies $f = g$ .