  In this paper, we introduce the category of brace triples in a braided
monoidal setting and prove that it is isomorphic to the category of s-Hopf
braces, which are a generalization of cocommutative Hopf braces. After that, we
obtain a categorical isomorphism between finite cocommutative Hopf braces and a
certain subcategory of cocommutative post-Hopf algebras, which suppose an
expansion to the braided monoidal setting of the equivalence obtained by Y. Li,
Y. Sheng and R. Tang for the category of vector spaces over a field
$\mathbb{K}$.
