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# identity functor

Let $\mathcal{C}$ be a category. The *identity functor* of $\mathcal{C}$ is the unique functor, written $I_{{\mathcal{C}}}$, such that for every object $A$ and every morphism $\alpha$ in $\mathcal{C}$, we have

$I_{{\mathcal{C}}}(A)=A\quad\mbox{ and }\quad I_{{\mathcal{C}}}(\alpha)=\alpha.$ |

To verify that $I_{{\mathcal{C}}}$ is indeed a functor, we note that $I_{{\mathcal{C}}}(1_{A})=1_{A}=1_{{I_{{\mathcal{C}}}(A)}}$, where $1_{A}$ is the identity morphism of $A$, and $I_{{\mathcal{C}}}(\alpha\circ\beta)=\alpha\circ\beta=I_{{\mathcal{C}}}(\alpha)% \circ I_{{\mathcal{C}}}(\beta)$.

For any functor $F:\mathcal{C}\to\mathcal{D}$, we have $F\circ I_{{\mathcal{C}}}=I_{{\mathcal{D}}}\circ F=F$.

Since every category gives rise to its unique identity functor, we can think of *the identity functor* $I$ as a (covariant) functor on Cat, the category of (small) categories. It is given by taking any category $\mathcal{C}$ to itself and any functor $F:\mathcal{C}\to\mathcal{D}$ to itself.

## Mathematics Subject Classification

18A05*no label found*18-00

*no label found*

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