cogenerating collection

This property refers to the existence of a cogenerating collection. A cogenerating 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 $h \circ f = h \circ g$ for every morphism $h : B \to Q$ with $Q \in S$, then $f = g$. Equivalently, the functor $$(\Hom(-,Q))_{Q \in S} : \C^{\op} \to (\Set^+)^S$$ is faithful. If $\C$ is locally essentially small and has products, this is also equivalent to the condition that the canonical morphism $$A \to \textstyle\prod_{Q\in S} \prod_{f\in\Hom(A,Q)} Q$$ is a monomorphism for every object $A$. Some authors say coseparating instead of cogenerating.The condition that $S$ is essentially small is important since, without it, every category would have a cogenerating 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 "cogenerating set". We did not choose "essentially small cogenerating collection" either since it is too cumbersome and diverges too much from the literature.