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 , p

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... are surjective submersions and an induced map Y 1 → Y 2 × Y 4 Y 3 , y → (p 12 (y), p 13 (y)) is a surjective submersion. A fibredfibred manifold has dimension (m 1 , m 2 , n 1 , n 2 ) if dim Y 1 = m 1 +m 2 +n 1 +n 2 , dim Y 2 = m 1 + m 2 , dim Y 3 = m 1 + n 1 , dim Y 4 = m 1 . For two fibredfibred manifolds Y, Y of the same dimension (m 1 , m 2 , n 1 , n 2 ), a morphism f : Y → Y is a quadruple of local diffeomorphisms f 1 : Y 1 → Y 1 , f 2 : Y 2 → 2010 Mathematics Subject Classification. 58A20, 58A32.