Let S be a commutative cancellative semigroup and Tq be a cofinal subsemigroup of S. Let Aq be a homomorphism of Tq into the semigroup of nonnegative real numbers under addition. We prove that Kobayashi's condition [2] is necessary and sufficient for h0 to be extended to 5. Further, we find a necessary and sufficient condition in order that the extension be unique. Related to this, the "boundedness condition" is introduced. For further study, several examples are given.