We continue the study of the category of functors ${ℱ}_{quad}$, associated to ₂-vector spaces equipped with a nondegenerate quadratic form, initiated in J. Pure Appl. Algebra 212 (2008) and Algebr. Geom. Topology 7 (2007). We define a filtration of the standard projective objects in ${ℱ}_{quad}$; this refines to give a decomposition into indecomposable factors of the first two standard projective objects in ${ℱ}_{quad}$: $P_{H₀}$ and $P_{H₁}$. As an application of these two decompositions, we give a complete description of the polynomial functors...