there is a braided $G$-crossed monoidal category $\mathcal{Z}_G({{\mathcal B}})$ such that the trivial component is $\mathcal{Z}({{\mathcal B}})$, the Drinfeld center of ${{\mathcal B}}$.

An important example comes from the theory of tensor categories. We show that, if ${{\mathcal D}}=\oplus_{g\in G} {{\mathcal D}}_g$ is a $G$-graded tensor category, and ${{\mathcal D}}_1={{\mathcal C}}$, there is an action of the group $G$ acts on ${}_{{{\mathcal C}}}{\mbox{\rm Mod\,}}$, the 2-category of representations of ${{\mathcal C}}$, and there is a biequivalence $$({}_{{{\mathcal C}}}{\mbox{\rm Mod\,}}^{{\rm{op}}})^G \simeq {}_{{{\mathcal D}}}{\mbox{\rm Mod\,}}.$$ The coherence theorem for group actions allows us to construct an associated strict braided crossed monoidal category and to prove that there is a monoidal equivalence between the center ${{\mathcal Z}}({{\mathcal B}}^G)$ of the equivariantization and the monoidal category of pseudonatural transformations of the forgetful pseudofunctor $\Phi:{{\mathcal B}}^G\to {{\mathcal B}}$. When applied this result to the 2-category $({}_{{{\mathcal C}}}{\mbox{\rm Mod\,}})^G $, we recover the results from [@GNN], on the center of graded tensor categories.

The contents of the paper are organized as follows. In Section \[Section:2-cat\] we recall the basics of 2-categories. For any pseudofunctor ${{\mathcal H}}:{{\mathcal B}}\to {{\mathcal B}}'$ we define the monoidal category ${{\mathcal Z}}({{\mathcal H}})$ of pseudonatural transformations $\eta:{{\mathcal H}}\to {{\mathcal H}}$. When ${{\mathcal 