nLab
identity morphism

The identity morphism, or simply identity, of an object $x$ in some category $C$ is the morphism $1_x:x\to x$, or ${id}_x:x\to x$, which acts as a two-sided identity for composition.

Given a small category $C$ with set of objects $C_0$ and set of morphisms $C_1$, the identity assigning function of $C$ is the function $i:C_0\to C_1$ that maps each object in $C_0$ to its identity morphism in $C_1$.

For the generalisation to an internal category $C$, see identity-assigning morphism.

In Set, the identity morphisms are the identity functions.