\begin{definition}[Definition:Restriction of Presheaf to Open Set]
Let $C$ be a category.
Let $X$ be a topological space.
Let $\FF$ be a $C$-valued presheaf on $X$.
Let $U \subset X$ be open.
The '''restriction''' of $\FF$ to $U$ is the restriction of the contravariant functor $\FF$ to the subcategory $U$.
\end{definition}