About Hopf braces and crossed products

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 $(A_1\bowtie H_1,A_2\mathrm{♯}{H}_2)$ to be Hopf braces, which are particular situations of the general problem described above, and we apply these results to study when the Drinfeld’s Double gives rise to a Hopf brace.