Lifting double linear vector fields to Weil like functors on double vector bundles

The complete description is given of the product preserving gauge bundle functors F on the category 2 − VB of double vector bundles in terms of the A F ‐bilinear maps⋄ F : U F × V F → W F , where A F are Weil algebras andU F , V F , W Fare finite dimensional (over R ) A F ‐modules. Then the so‐called “iteration” problem is solved (i.e. ⋄ F ′∘ Fby means of ⋄ F and ⋄ F ′is computed). That any two product preserving gauge bundle functors on 2 − VB commute is derived. All gauge natural operators lifting double linear vector fields to product preserving gauge bundle functors F are completely described.