For a monoidal category , a -enriched category has hom-objects in , composition morphisms , and unit morphisms satisfying associativity and unit laws.
A closed symmetric monoidal category enriches over itself using its internal hom objects. Composition is adjoint to the composite of evaluation morphisms.
In a category whose hom-sets are posets, is left adjoint to when and . Left adjoints are closed under identities and composition.
In the inclusion-ordered category of sets and relations, a relation has a right adjoint exactly when it is the graph of a total single-valued function; its right adjoint is the converse relation.
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The term "enriched category" typically arises in the context of category theory, a branch of mathematics that deals with abstract structures and relationships between them. In general, a category consists of objects and morphisms (arrows) that represent relationships between those objects. An **enriched category** expands this concept by allowing the hom-sets (the sets of morphisms between objects) to take values in a more general structure than merely sets.