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 $C_1$.