nLab
semipresheaf

Definition

A semipresheaf on a semicategory $C$ is a semifunctor

$$F:C^{\mathrm{op}}\to \mathrm{Set}$$F : C^{op} \to Set

from the opposite semicategory $C^{\mathrm{op}}$ of $C$ to the category Set of sets.

The category of semipresheaves on $C$, usually denoted $[C^{\mathrm{op}},\mathrm{Set}{]}_{\mathrm{semi}}$, or just $[C^{\mathrm{op}},\mathrm{Set}]$ when the context is clear has:

• semifunctors $F:C^{\mathrm{op}}\to \mathrm{Set}$ as objects;

• natural transformations between such semifunctors as morphisms.