1.NounAn 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
S
e
t
{\displaystyle \mathbf {Set} }
; namely, that of as a function set or internal hom-set.; The pair
Z
Y
,
eval
:
Z
Y
×
Y
→
Z
{\displaystyle Z^{Y},{\mbox{eval}}:Z^{Y}\times Y\rightarrow Z}
is the terminal object of the comma category
(
−
×
Y
)
↓
Z
{\displaystyle (-\times Y)\downarrow Z}
. Therefore the exponential object is a kind of universal morphism."