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Combinatorial structures which compose and decompose give rise to Hopfmonoids in Joyal's category of species. The Hadamard product of two Hopfmonoids is another Hopf monoid. We prove two main results regarding freeness ofHadamard products. The first one states that if one factor is connected and theother is free as a monoid, their Hadamard product is free (and connected). Thesecond provides an explicit basis for the Hadamard product when both factorsare free.  The first main result is obtained by showing the existence of a one-parameterdeformation of the comonoid structure and appealing to a rigidity result ofLoday and Ronco which applies when the parameter is set to zero. To obtain thesecond result, we introduce an operation on species which is intertwined by thefree monoid functor with the Hadamard product. As an application of the firstresult, we deduce that the dimension sequence of a connected Hopf monoidsatisfies the following condition: except for the first, all coefficients ofthe reciprocal of its generating function are nonpositive.

On the Hadamard product of Hopf monoids