Geodesic
Non-existence of natural operators transforming connections on $Y \to M$ into connections on $FY\to Y$
Archivum mathematicum, Tome 41 (2005) no. 1, pp. 1-4.

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Under some weak assumptions on a bundle functor $F:FM_{m,n}\rightarrow FM$ we prove that there is no $FM_{m,n}$-natural operator transforming connections on $Y\rightarrow M$ into connections on $FY\rightarrow Y$.
Classification : 58A05, 58A20, 58A32
Keywords: bundle functors; natural bundles; natural operators; connections 
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     title = {Non-existence of natural operators transforming connections on $Y \to M$ into connections on $FY\to Y$},
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Mikulski, W. M. Non-existence of natural operators transforming connections on $Y \to M$ into connections on $FY\to Y$. Archivum mathematicum, Tome 41 (2005) no. 1, pp. 1-4. http://geodesic.mathdoc.fr/item/ARM_2005__41_1_a0/