  Let ${\mathbf U}^-_q$ be the negative part of the quantum enveloping algebra
associated to a simply laced Kac-Moody Lie algebra ${\mathfrak g}$, and
$\underline{\mathbf U}^-_q$ the algebra corresponding to the fixed point
subalgebra of ${\mathfrak g}$ obtained from a diagram automorphism $\sigma$ on
${\mathfrak g}$. Let ${\mathbf B}^{\sigma}$ be the set of $\sigma$-fixed
elements in the canonical basis of ${\mathbf U}_q^-$, and $\underline{\mathbf
B}$ the canonical basis of $\underline{\mathbf U}_q^-$. Lusztig proved that
there exists a canonical bijection ${\mathbf B}^{\sigma} \simeq
\underline{\mathbf B}$ based on his geometric construction of canonical bases.
In this paper, we prove (the signed bases version of) this fact, in the case
where ${\mathfrak g}$ is finite or affine type, in an elementary way, in the
sense that we don't appeal to the geometric theory of canonical bases nor
Kashiwara's theory of crystal bases. We also discuss the correspondence between
PBW-bases, by using a new type of PBW-bases of ${\mathbf U}_q^-$ obtained by
Muthiah-Tingley, which is a generalization of PBW-bases constructed by
Beck-Nakajima.
