Presheaf and Sheaf

Presheaf

A presheaf on a category $\mathsf{C}$ is a functor \newcommand{\Set}{\mathsf{Set}}F\colon \mathsf{C}^{\text{op}}\to\Set.
In particular, if $\mathsf{C}$ is a topological space $X$ (i.e., the category of open sets of $X$), then a presheaf on $X$ is a functor \mathsf{Open}(X)^{\text{op}}\to\Set.