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Hecke-Hopf algebras were defined by A. Berenstein and D. Kazhdan. We give anexplicit presentation of an Hecke-Hopf algebra when the parameter $m_{ij},$associated to any two distinct vertices $i$ and $j$ in the presentation of aCoxeter group, equals $4,$ $5$ or $6$. As an application, we give a proof of aconjecture of Berenstein and Kazhdan when the Coxeter group is crystallographicand non-simply-laced. As another application, we show that another conjectureof Berenstein and Kazhdan holds when $m_{ij},$ associated to any two distinctvertices $i$ and $j,$ equals $4$ and that the conjecture does not hold whensome $m_{ij}$ equals $6$ by giving a counterexample to it.

On the presentation of Hecke-Hopf algebras for non-simply-laced type