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The exponential object (with evaluation morphism ) indexes morphisms from Y to Z in a universal way, which means that for a family of morphisms from Y to Z indexed by X (with evaluation morphism ), there is a unique morphism , called the transpose of g, such that gfactors through with cofactor .
(category theory) An object which indexes a family of arrows between two given objects in a universal way, meaning that any other indexed family of arrows between the same given pair of objects must factor uniquely through this universally-indexed family of arrows.
An exponential object generalizes its interpretation in category ; namely, that of as a function set or internal hom-set.
The pair is the terminal object of the comma category . Therefore the exponential object is a kind of universal morphism.
