Centers of Algebras in Monoidal 2-Categories

We introduce the left, right, and full center of an algebra in a monoidal 2-category and prove their Morita invariance. This categorifies Davydov's theory of centers of algebras in monoidal categories, and specializes to give a uniform, structural account of the Drinfeld center of a fusion category, the crossed braided Drinfeld center of a fusion category graded by a finite group, and the center of a central module monoidal category. Along the way, we develop a theory of 2-adjunctions between monoidal 2-categories and a base-change construction for module 2-categories.