Factorization of homotopies of nanophrases

Homotopy on nanophrases is an equivalence relation defined using some data called a homotopy data triple. We define a product on homotopy data triples. We show that any homotopy data triple can be factorized into a product of prime homotopy data triples and this factorization is unique up to isomorphism and order. If a homotopy data triple is composite, we show that equivalence of nanophrases under the corresponding homotopy can be calculated just by using the homotopies given by its prime factors.