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

Assume every morphism of $\mathcal{C}$ is monic. For a representable presheaf $F=\mathcal{C}(-,A)$, the category $(1\downarrow F)$ is the category of elements, equivalently the slice $\mathcal{C}/A$. A morphism from $f:B\to A$ to $g:C\to A$ is an $h:B\to C$ satisfying $gh=f$. Since $g$ is monic, there is at most one such $h$. Thus $(1\downarrow F)$ is a preorder.