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.