On lifts of projectable-projectable classical linear connections to the cotangent bundle

We describe all ${\mathcal{F}}^2{\mathcal{M}}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝Q{T}^{\ast }$ transforming projectable-projectable classical torsion-free linear connections $\mathrm{\nabla }$ on fibred-fibred manifolds $Y$ into classical linear connections $D(\mathrm{\nabla })$ on cotangent bundles $T^{\ast }Y$ of $Y$. We show that this problem can be reduced to finding ${\mathcal{F}}^2{\mathcal{M}}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝(T^{\ast },{\otimes }^p{T}^{\ast }\otimes {\otimes }^{q}T)$ for $p=2$, $q=1$ and $p=3$, $q=0$.