extremal generating collection

This property refers to the existence of an extremal generating collection. A collection of objects $S \subseteq \Ob(\C)$ in a category $\C$ is called an extremal generating collection if it is a generating collection and for every morphism $f : A \to B$, $f$ is an isomorphism if and only if for every object $G \in S$ we have $f \circ {-} : \Hom(G, A) \to \Hom(G, B)$ is a bijection. Equivalently, the functor $$(\Hom(G,-))_{G \in S} : \C \to (\Set^+)^S$$ is faithful and conservative. If $\C$ is locally essentially small and has coproducts, it 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 extremal epimorphism for every object $A$, explaining the terminology (see Prop. 5.3 at the nLab). The term "extremal generating set" is more common, but within our foundations, the collection $S$ is not necessarily a set; it is just isomorphic to a set.