In the paper the authors study the prolongation of projectable tangent valued forms on a fibered manifold \(p: Y\to M\) to fibred bundle \(T^\mu Y\), where \(T^\mu \) is a product preserving bundle functor induced by W. M. Mikulski [Arch. Math., Brno 32, No. 4, 307–316 (1996; Zbl 0881.58002)]. The complete lift of the Frölicher-Nijenhuis bracket is also studied.