extremal cogenerating collection

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