The Stacks project

Definition 6.5.1. Let $X$ be a topological space. Let $\mathcal{C}$ be a category.

1. A presheaf $\mathcal{F}$ on $X$ with values in $\mathcal{C}$ is given by a rule which assigns to every open $U \subset X$ an object $\mathcal{F}(U)$ of $\mathcal{C}$ and to each inclusion $V \subset U$ a morphism $\rho _ V^ U : \mathcal{F}(U) \to \mathcal{F}(V)$ in $\mathcal{C}$ such that whenever $W \subset V \subset U$ we have $\rho _ W^ U = \rho _ W^ V \circ \rho _ V^ U$.

2. A morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves with value in $\mathcal{C}$ is given by a morphism $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ in $\mathcal{C}$ compatible with restriction morphisms.