A schema algebra comprises operations on database schemata for a given data model. Such algebras are useful in database design as well as in schema integration. In this article we address the necessary theoretical underpinnings by introducing a novel notion of conceptual schema morphism that captures at the same time the conceptual schema and its semantics by means of the set of valid instances. This leads to a category of schemata that is finitely complete and co-complete. This is the basis for a notion of completeness of schema algebras, if it captures all universal constructions in the category of schemata. We exemplify this notion of completeness for a recently introduced particular schema algebra.