Definition:Identity Functor

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Let $\mathbf C$ be a metacategory.

The identity functor on $\mathbf C$ is the functor $\operatorname{id}_{\mathbf C}: \mathbf C \to \mathbf C$ defined by:

For all objects $C$ of $\mathbf C$: $\hskip{2.9cm} \operatorname{id}_{\mathbf C} C := C$
For all morphisms $f: C_1 \to C_2$ of $\mathbf C$: $\quad \operatorname{id}_{\mathbf C} f := f$

That $\operatorname{id}_{\mathbf C}$ constitutes a functor is shown on Identity Functor is Functor.