Let’s look more deeply at how \\(\mathcal{D}^\mathcal{C}\\) is a category.

1. Figure out how to compose natural transformations. (Hint: an expert tells you “for each object \\(c \in \mathcal{C}\\), compose the \\(c\\)-components”.)

2. Propose an identity natural transformation on any object \\(F \in \mathcal{D}^\mathcal{C}\\) , and check that it is unital.

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**Definition 3.44**

Let \\(\mathcal{C}\\) and \\(\mathcal{D}\\) be categories.

We denote by \\(\mathcal{D}^\mathcal{C}\\) the category whose objects are functors \\(F : \mathcal{C} \rightarrow \mathcal{D}\\) and whose morphisms \\(\mathcal{D}^\mathcal{C}(F, G)\\) are the natural transformations \\( \alpha : F \rightarrow G\\).

This category \\(\mathcal{D}^\mathcal{C}\\) is called the \\(\textit{functor category}\\).

1. Figure out how to compose natural transformations. (Hint: an expert tells you “for each object \\(c \in \mathcal{C}\\), compose the \\(c\\)-components”.)

2. Propose an identity natural transformation on any object \\(F \in \mathcal{D}^\mathcal{C}\\) , and check that it is unital.

[Previous](https://forum.azimuthproject.org/discussion/2150)

[Next](https://forum.azimuthproject.org/discussion/2152)

**Definition 3.44**

Let \\(\mathcal{C}\\) and \\(\mathcal{D}\\) be categories.

We denote by \\(\mathcal{D}^\mathcal{C}\\) the category whose objects are functors \\(F : \mathcal{C} \rightarrow \mathcal{D}\\) and whose morphisms \\(\mathcal{D}^\mathcal{C}(F, G)\\) are the natural transformations \\( \alpha : F \rightarrow G\\).

This category \\(\mathcal{D}^\mathcal{C}\\) is called the \\(\textit{functor category}\\).