The identity natural transformation $\text{id}_{F}\colon F\Longrightarrow F$ of $F$ is the natural transformation consisting of the collection

\[ \webleft\{ \text{id}_{F\webleft (A\webright )}\colon F\webleft (A\webright )\to F\webleft (A\webright )\webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )}. \]

The naturality condition for $\text{id}_{F}$ is the requirement that, for each morphism $f\colon A\to B$ of $\mathcal{C}$, the diagram

commutes, which follows from unitality of the composition of $\mathcal{C}$.


Noticed something off, or have any comments? Feel free to reach out!


You can also use the contact form below: