nLab
identity function

Given a set $S$, the identity function on $S$ is the function ${\mathrm{id}}_S:S\to S$ that maps any element $x$ of $S$ to itself:

$${\mathrm{id}}_S=(x\mapsto x)=\lambda x.x;$$id_S = (x \mapsto x) = \lambda x.\; x ;

or equivalently,

$${\mathrm{id}}_S(x)=x.$$id_S(x) = x .

The identity functions are the identity morphisms in the category Set of sets.

More generally, in any concrete category, the identity morphism of each object is given by the identity function on its underlying set.