Pullback of lie algebra and lie group bundles, and their homotopy invariance

We study the pullback Lie algebra (group) bundle of a Lie algebra (group) bundle and show that the Lie algebra bundle of the pullback of a Lie group bundle $mathfrak{G}$ is isomorphic to the pullback of the Lie algebra bundle of $mathfrak{G}$. Then, using the notion of Lie connection on a Lie algebra bundle, we show that the pullbacks of a Lie algebra bundle $xi$ over a smooth manifold $M$ with respect to two smooth homotopic functions $f_0 , f_1 : N rightarrow M$ are isomorphic to Lie algebra bundles over $N$.