Center construction for group-crossed tensor categories

We define the notion of the $(G,\mathrm{\Gamma })$-crossed center of a $(G,\mathrm{\Gamma })$-crossed tensor category in the sense of Natale. We show that the $(G,\mathrm{\Gamma })$-crossed center is a $(G\bowtie \mathrm{\Gamma },G\times \mathrm{\Gamma })$-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.