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This paper presents the semantics of database mappings in the relational database (DB) category, based on the power-view monad T and monadic algebras. The semantics can be interpreted as a computational model of view-based mappings between databases, where each query (view-mapping) can be seen as a program, so that we can use the paradigm 'from values to computations'. The objects in this category are the database-instances. The morphisms of such DB category are used in order to express the semantics of view-based global and local as view (GLAV) mappings between relational databases such as those used in data integration systems. Consequently, the semantics of database mappings in this DB category are defined based on the power-view monad T and the Kleisli category for databases, which can be 'internalised' in this basic DB category.