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It is proved in \cite{BP} (arXiv:1108.2717) that the category of relative3-dimensional cobordisms $\cal Cob^{2+1}$ is equivalent to the universalalgebraic category $\overline{\overline{\cal H}}{}^r$ generated by a Hopfalgebra object. A different algebraic category $\overline{\cal Alg}$ generatedby a Hopf algebra object is defined in \cite{AS} and it conjectured to beequivalent to $\cal Cob^{2+1}$ as well. We prove that there exists a functor$\overline{\cal Alg}\to \overline{\overline{\cal H}}{}^r$, and use it topresent an alternative set of axioms for $\overline{\overline{\cal H}}{}^r$.

On the algebraic characterization of the category of 3-dimensional cobordisms