nLab identity natural transformation

Definition

The identity natural transformation on a functor $F: C \to D$ is the natural transformation $id_F: F \to F$ that maps each object $x$ of $C$ to the identity morphism $id_{F(x)}$ in $D$.

The identity natural transformations are themselves the identity morphisms for vertical composition of natural transformations in the functor category $D^C$ and in the 2-category Cat.

One must be aware that when we say that a natural transformation $\alpha_{A}:F(A) \rightarrow G(A)$ is the identity, it doesn’t mean only that $F(A)=G(A)$ for every object $A$ but also that $F(f)=G(f)$ for every morphism $f$ ie. that the two functors $F$ and $G$ are equal.

Not taking care of this can lead to redundant definitions as for permutative categories.

Related concepts

• equality, equivalence

• identity function, identity morphism

• identity functor

• identity type

• identity of indiscernibles