# Definition:Functor Category

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## Definition

Let $C$ and $D$ be categories.

The **functor category** $\operatorname{Funct}(C, D)$ is the category with:

Objects: | covariant functors $C \to D$ | |

Morphisms: | natural transformations | |

Composition: | vertical composition of natural transformations | |

Identity morphisms: | identity natural transformations |

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## Also denoted as

The **functor category** is also denoted $\operatorname{Fun}(C, D)$, $[C, D]$ or $D^C$, in analogy to the set of all mappings.