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We describe all F 2 Mm 1 ,m 2 ,n 1 ,n 2 -natural operators D : Q τ proj-proj QT * transforming projectable-projectable classical torsion-free linear connections ∇ on fibred-fibred manifolds Y into classical linear connections D(∇) on cotangent bundles T * Y of Y . We show that this problem can be reduced to finding F 2 Mm 1 ,m 2 ,n 1 ,n 2 -natural operators D : Q τ proj-proj (T * , ⊗ p T * ⊗ ⊗ q T ) for p = 2, q = 1 and p = 3, q = 0. Basic definitions and examples. maps p 12 , p 13 , p 24 , pdoi:10.2478/v10062-012-0017-x fatcat:wlyomj7umvcbrhybsndemwcbue