generating collection

This property refers to the existence of a generating collection. A generating collection of a category $\C$ is a collection of objects $S \subseteq \Ob(\C)$ that is essentially small (i.e., isomorphic to a set) and has the following property: if parallel morphisms $f,g : A \rightrightarrows B$ satisfy $f \circ h = g \circ h$ for every morphism $h : G \to A$ with $G \in S$, then $f = g$. Equivalently, the functor $$(\Hom(G,-))_{G \in S} : \C \to (\Set^+)^S$$ is faithful. If $\C$ is locally essentially small and has coproducts, this is also equivalent to the condition that the canonical morphism $$\textstyle\bigsqcup_{G\in S} \bigsqcup_{f\in\Hom(G,A)} G \to A$$ is an epimorphism for every object $A$.
The condition that $S$ is essentially small is important since, without it, every category would have a generating collection, namely its collection of all objects. Notice that $S$ is not assumed to be a set "on the nose", since its objects do not need to be sets; see our foundations for more background. This is why we have not adopted the more common terminology of a "generating set". We did not choose "essentially small generating collection" either since it is too cumbersome and diverges too much from the literature.