Definition:Sheaf on Topological Space/Definition 4

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $\mathbf C$ be a category.

A $\mathbf C$-valued sheaf $\FF$ on $T$ is a $\mathbf C$-valued presheaf such that for all objects $X$ of $\mathbf C$ the presheaf of sets $\FF_X$ defined by
 * $\FF_X(U) := \mathrm{Hom}_{\mathbf C}(X, \FF(U))$

is a sheaf of sets on $T$.

Also see

 * Equivalence of Definitions of Sheaf on Topological Space