The paper gives an operator algebras model for the conditional monotone
independence, introduced by T. Hasebe. The construction is used to prove an
embedding result for the N. Muraki's monotone product of C*-algebras. Also, the
formulas from the definition of conditional monotone independence are used to
define the monotone product of maps which is shown to preserve complete
positivity, a similar to the results from the case of free products.