  Freed, Hopkins and Teleman constructed an isomorphism between twisted
equivariant K-theory of compact Lie group $G$ and the "Verlinde ring" of the
loop group of $G$. We call this isomorphism FHT isomorphism. However, it does
not hold naturality with respect to group homomorphisms. We construct two
"quasi functors" $t.e.K$ (a modification of twisted equivariant K-theory) and
$RL$ (a modification of representation group of loop groups) so that FHT
isomorphism is natural transformation between two "quasi functors" for tori,
that is, we construct two "induced homomorphisms" of the "quasi functors"
$t.e.K$ and $RL$ for a group homomorphism whose tangent map is injective
between two tori. In fact, we construct another quasi functor $char$ and verify
that three quasi functors are naturally isomorphic.
  Moreover, we extend the quasi functor $t.e.K$ and $char$ to compact connected
Lie group with torsion-free fundamental group and group homomorphism satisfying
"the decomposable condition", and verify that they are isomorphic. This is a
generalization of a result in [FHT1].
