Past exam of the introduction to schemes course of the University of Oxford / 2023 / 1 / a / iii / Solution

A presheaf $F:{\mathsf{Sch}}^{\mathrm{op}}\to \mathsf{Set}$ is a Zariski sheaf when every Zariski cover $(U_i\to T)$ gives an equalizer diagramThus compatible local elements glue to a unique element of $F(T)$. Compatibility means that the two restrictions agree on every intersection. This includes the empty Zariski cover of the empty scheme, so $F(\varnothing )$ is a singleton.