Cogenerating set ($(Q_i{)}_{i\in I}$)

A cogenerating set is a set-indexed family $(Q_i{)}_{i\in I}$ such that whenever $f\ne g:A\to B$, there is a morphism $q:B\to Q_i$ with $qf\ne qg$. It generalizes a single coseparator. In a locally small category with the needed products in a category, it gives an evaluation embedding into cogenerator products.

For a small cogenerating family $(Q_i)$ in a locally small category, the evaluation arrow $C\to {\prod }_{i,q\in \mathcal{C}(C,Q_i)}{Q}_i$ is a monomorphism whenever that small product in a category exists. Equality after all its projections is equality after every map to a cogenerator, hence equality of the original parallel morphisms.