Transitive Lie algebroids. Categorical point of view

In this paper, the functorial property of the inverse image for transitive Lie algebroids is proved and also there is proved the functorial property for all objects that are necessary for building transitive Lie algebroids due to K. Mackenzie – bundles $L$ of finite-dimensional Lie algebras, covariant connections of derivations $\nabla$, associated differential $2$-dimensional forms $\Omega$ with values in the bundle $L$, couplings, and the Mackenzie obstructions. On the base of the functorial properties, a final object for the structure of transitive Lie prealgebroid and for the universal cohomology class inducing the Mackenzie obstruction can be constructed.