The core of a model theory for generic schema management is developed. This theory has two distinctive features: it applies to a variety of categories of schemas, and it applies to transformations of both the schema structure and its integrity constraints. A subtle problem of schema integration is considered in its general form, not bound to any particular category of schemas. The proposed solution, as well as the overall theory, is based entirely on schema morphisms that carry both structural and semantic properties. Duality results that apply to the schema and the data levels are established. These results lead to the main contribution of this paper: a formal schema and data management framework for generic schema management. Implications of this theory are established that apply to integrity problems in schema integration. The theory is illustrated by a particular category of schemas with object-oriented features along with typical database integrity constraints.