  We classify isomorphic classes of the homomorphisms of a root system $\Xi$ to
a root system $\Sigma$ which do not change Cartan integers. We examine several
types of isomorphic classes defined by the Weyl group of $\Sigma$, that of
$\Xi$ and the automorphisms of $\Sigma$ or $\Xi$ etc. We also distinguish the
subsystem generated by a subset of a fundamental system. We introduce the
concept of the dual pair for root systems which helps to study the action of
the outer automorphism of $\Xi$ on the homomorphisms.
