The vertical prolongation of the projectable connections

We prove that any first order ${\mathcal{F}}_2{\mathcal{M}}_{m_1,m_2,n_1,n_2}$-natural operator transforming projectable general connections on an $(m_1,m_2,n_1,n_2)$-dimensional fibred-fibred manifold $p=(p,p):(p_Y:Y\to Y)\to (p_M:M\to M)$ into general connections on the vertical prolongation $VY\to M$ of $p:Y\to M$ is the restriction of the (rather well-known) vertical prolongation operator $\mathcal{V}$ lifting general connections $\bar{\mathrm{\Gamma }}$ on a fibred manifold $Y\to M$ into $\mathcal{V}\bar{\mathrm{\Gamma }}$ (the vertical prolongation of $\bar{\mathrm{\Gamma }}$) on $VY\to M$.