[[Natural transformation]]
# Identity natural transformation

The **identity natural transformation** of a [[functor]] $F : \cat C \to \cat D$ is the natural transformation $\id_{F} \in \cat D^{\cat C}(F,F)$ whose component for every object $x \in \cat C$ is the identity morphism #m/def/cat 
$$
\begin{align*}
(\id_{F})_{x} = \id_{Fx} = F(\id_{x})
\end{align*}
$$
The form the identity morphisms in the [[functor category]] $\cat D^\cat{C}$.

## Properties

- [[Identity as a natural transformation]]

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#state/tidy | #lang/en | #SemBr