The present article represents a step forward in the study of the following problem: If \(\mathbb {A}=(A_{1},A_{2})\) and \(\mathbb {H}=(H_{1},H_{2})\) are Hopf braces in a symmetric monoidal category C such that \((A_{1},H_{1})\) and \((A_{2},H_{2})\) are matched pairs of Hopf algebras, then we want to know under what conditions the pair \((A_{1}\bowtie H_{1},A_{2}\bowtie H_{2})\) constitutes a new Hopf brace. We find such conditions for the pairs \((A_{1}\otimes H_{1},A_{2}\bowtie H_{2})\) and \((A_{1}\bowtie H_{1},A_{2}\sharp H_{2})\) to be Hopf braces, which are particular situations of the general problem described above, analyzing when these are cocommutative, leading to solutions to the Quantum Yang-Baxter equation. These results are applied to study when the Drinfeld’s Double gives rise to a Hopf brace.
