  Transitive Lie algebroids have specific properties that allow to look at the
transitive Lie algebroid as an element of the object of a homotopy functor.
Roughly speaking each transitive Lie algebroids can be described as a vector
bundle over the tangent bundle of the manifold which is endowed with additional
structures. Therefore transitive Lie algebroids admits a construction of
inverse image generated by a smooth mapping of smooth manifolds. Due to to
K.Mackenzie (2005) the construction can be managed as a homotopy functor
$\mathcal{TLA}_{\rg}$ from category of smooth manifolds to the transitive Lie
algebroids. The functor $\mathcal{TLA}_{\rg}$ associates with each smooth
manifold $M$ the set $\mathcal{TLA}_{\rg}(M)$ of all transitive algebroids with
fixed structural finite dimensional Lie algebra $\rg$. Hence one can construct
a classifying space $\cB_{\rg}$ such that the family of all transitive Lie
algebroids with fixed Lie algebra $\rg$ over the manifold $M$ has one-to-one
correspondence with the family of homotopy classes of continuous maps
$[M,\cB_{\rg}]$: $\mathcal{TLA}_{\rg}(M)\approx [M,\cB_{\rg}].$
  It allows to describe characteristic classes of transitive Lie algebroids
from the point of view a natural transformation of functors similar to the
classical abstract characteristic classes for vector bundles and to compare
them with that derived from the Chern-Weil homomorphism by J.Kubarski. As a
matter of fact we show that the Chern-Weil homomorphism does not cover all
characteristic classes from categorical point of view.
