Then, if we are given a connection in terms of a G-invariant subbundle of $TP$, we could express its inclusion map $H \to TP$ as a map $H \to TM$ together with a certain 2-cell between the composite of the canonical map $H \to $ (the universal TG-bundle composed with the unique map to the terminal object) and T(p) composed with $H \to TM$. I was thinking of using the fact that we can obtain $P$ as the weak pullback of the classifying map p along the universal principal G-bundle $* \to $, and then using that the tangent functor preserves finite weak limits. If P is equipped with a connection, how does this fit in this picture? Then by Yoneda, this corresponds to a smooth map $p:M \to $, where $$ is the differentiable stack associated to G. Suppose that P is a principal G-bundle over M.
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