Center construction for group-crossed tensor categories

We define the notion of the $(G,\Gamma)$-crossed center of a $(G,\Gamma)$-crossed tensor category in the sense of Natale. We show that the $(G, \Gamma)$-crossed center is a $(G \bowtie \Gamma, G \times \Gamma)$-braided tensor category. This construction generalizes the graded center construction for graded tensor categories and the equivariant center construction for tensor categories with group actions.