In this paper, we define the functor category Fquad associated to vector spaces over the field with two elements, F_2, equipped with a quadratic form. We show the existence of a fully-faithful, exact functor \iota: \F \rightarrow Fquad, which preserves simple objects, where \F is the category of functors from the category of finite dimensional F_2-vector spaces to the category of all F_2-vector spaces. We define the subcategory Fiso of Fquad, which is equivalent to the product of the categories of modules over the orthogonal groups; the inclusion is a fully-faithful functor \kappa: Fiso \rightarrow Fquad which preserves simple objects.