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.