nLab
identity natural transformation

The identity natural transformation on a functor $F:C\to D$ is the natural transformation ${\mathrm{id}}_F:F\to F$ that maps each object $x$ of $C$ to the identity morphism ${\mathrm{id}}_{F(x)}$ in $D$. The identity natural transformations are the identities for vertical composition of natural transformations in the functor category $D^C$ and in the 2-category Cat.