The identity natural transformation $\text{id}_{F}\colon F\Rightarrow F$ of $F$ is the natural transformation consisting of the collection
\[ \webleft\{ \webleft (\text{id}_{F}\webright )_{A}\colon F\webleft (A\webright )\to F\webleft (A\webright )\webright\} _{A\in \text{Obj}\webleft (\mathcal{C}\webright )} \]
defined by
\[ \webleft (\text{id}_{F}\webright )_{A}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{F\webleft (A\webright )} \]
for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$.
