  In this paper, we mainly prove a theorem with a corollary establishing two
characterizations of the Calabi composition of hyperbolic hyperspheres, where
the second characterization (i.e., the corollary) has been given via a dual
correspondence theorem earlier but now we would like to use a very direct
method. Note that Z.J. Hu, H.Z. Li and L. Vrancken also gave a characterization
of the $2$-factor Calabi composition in a different manner.
