Hopf Galois Extension in Braided Tensor Categories

The relation between crossed product and $H$-Galois extension in braided tensor category $\mathcal{C}$ with equivalisers and coequivalisers is established. That is, it is shown that if there exist an equivaliser and a coequivaliser for any two morphisms in $\mathcal{C}$, then A = B #_\sigma H is a crossed product algebra if and only if the extension $A\mathrm{/}B$ is Galois, the canonical epic $q: A\otimes A \to A\otimes_B A$issplitand$A$isisomorphicasleft$B$-modules and right $H$-comodules to $B\otimes H$ in $\mathcal{C}$.