Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 119 / 1 / b / Solution

Assume (b). A coproduct in a category of monofunctors in $[\mathcal{C},\mathbf{Set}]$ is again a monofunctor: at an object it is the disjoint union of the values, and every arrow acts within the same summand by an injective function. Distinct summands cannot become identified, and injectivity within a summand follows from its monofunctor property.

Apply this to the coproduct of representable functors constructed in part (a). Its canonical map to $F$ is a pointwise epimorphism in a functor category. Thus every $F$ is an epimorphic image of a monofunctor, proving