Evaluation map of an exponential object
= Evaluation map of an exponential object
{title2=$\operatorname{ev}:Y^X\times X\to Y$}
The evaluation map is the counit representing application in an <exponential object>. For every $Z$, composition with evaluation gives the natural bijection $\operatorname{Hom}(Z,Y^X)\cong\operatorname{Hom}(Z\times X,Y)$. Its inverse is <currying>. Evaluation must be a morphism in the ambient category, such as an <equivariant map> in a category of <group> actions.