nLab
identity-assigning morphism

Definition

Given an internal category $C$ with object of objects $C_0$ and object of morphisms $C_1$, the identity-assigning morphism of $C$ is the morphism $i:C_0\to C_1$ that is part of the definition of internal category.

This generalises the identity-assigning function of a small category $C$. Given such a small category 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 simplicial sets and simplicial objects, the identity-assigning morphisms are the degeneracy map?s .