Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 1 / c / ii / Solution

Assume every category of elements of a representable presheaf is a preorder. LetA disjoint union of preorders is a preorder. The category-of-elements projection $P\to \mathcal{C}$ is a discrete fibration. It is surjective on objects because $B\in \mathcal{C}$ is the image of the object $1_B:B\to B$ in the summand indexed by $B$.