The ability to cast values between related types is a leitmotiv of many flavors of dependent type theory, such as observational type theories, subtyping, or cast calculi for gradual typing. These casts all exhibit a common structural behavior that boils down to the pervasive functoriality of type formers. We propose and extensively study a type theory, called AdapTT, which makes systematic and precise this idea of functorial type formers, with respect to an abstract notion of adapters relating types. Leveraging descriptions for functorial inductive types in AdapTT, we derive structural laws for type casts on general inductive type formers.