  In this paper, we mainly give some equivalent characterisations of Hopf
braces, show that the category $\mathcal{CB}(A)$ of Hopf braces is equivalent
to the category $\mathcal{C}(A)$ of bijective 1-cocycles, and prove that the
category $\mathcal{CB}(A)$ of Hopf braces is also equivalent to the category
$\mathcal{M}(A)$ of Hopf matched pairs. Moreover, we construct many more Hopf
braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel'd
doubles of finite dimensional Hopf algebras, and give a sufficient and
necessary condition for a given bicrossed coproduct $A\bowtie H$ to be a Hopf
brace if $A$ or $H$ is a Hopf brace.
