  We examine functorial and homotopy properties of the exotic characteristic
homomorphism in the category of Lie algebroids which was lastly obtained by the
authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B
$\subset$ A are regular Lie algebroids, both over the same regular foliated
manifold (M,F), and $\nabla$ is a flat L-connection in A, where L is an
arbitrary Lie algebroid over M. The Rigidity Theorem (i.e. the independence
from the choice of homotopic Lie subalgebroids of B) is obtained. The exotic
characteristic homomorphism is factorized by one (called universal) obtained
for a pair of regular Lie algebroids. We raise the issue of injectivity of the
universal homomorphism and establish injectivity for special cases. Here the
Koszul homomorphism for pairs of isotropy Lie algebras plays a major role.
