Opposite brace triples form a category isomorphic to Hopf braces under cocommutativity and to matched pairs of Hopf algebras in fixed subcategories.
Mac Lane,Categories for the working mathematician, Graduate Texts in Mathematics, Vol
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Opposite brace triples, Hopf braces and matched pairs of Hopf algebras
Opposite brace triples form a category isomorphic to Hopf braces under cocommutativity and to matched pairs of Hopf algebras in fixed subcategories.