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}\$$.