The exponential objectZY — which may be pronounced "Z by Y" — indexes a family of arrows from Y to Z in a universal way, which means that for any X-indexed family of arrows g from Y to Z, g must factor through eval by means of an unique other arrow which is the product of the transpose of g and the identity of Y.
(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.