  Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas
to construct the perverse sheaves over the corresponding quiver and the
functions of irreducible components over the corresponding preprojective
algebra respectively. In the present article, we prove that these two inductive
algorithms of Lusztig coincide. Consequently we can define two colored graphs
and prove that they are isomorhic. This result finishes the statement that
Lusztig's functions of irreducible components are basis of the enveloping
algebra and deduces the crystal structure (in the sense of Kashiwara-Saito)
from the semicanonical basis directly inside Lusztig's convolution algebra of
the preprojective algebra. As an application, we prove that the transition
matrix between the canonical basis and the semicanonical basis is upper
triangular with all diagonal entries equal to 1.
