Functor category

Remarks

The composite $\beta \circ \alpha$ is natural, because both of its naturality squares can be pasted together. It is also written $\mathcal{D}^{\mathcal{C}}$ or $\mathrm{Fun}(\mathcal{C}, \mathcal{D})$. Smallness of $\mathcal{C}$ ensures that, when $\mathcal{D}$ is locally small, the natural transformations between two functors form a set. Example: let $G$ be a group, seen as a category with one object $\ast$ and with the elements of $G$ as morphisms. A functor $G \to \mathbf{Set}$ is a set $X = F(\ast)$ with a group action $g \cdot x = F(g)(x)$, and a natural transformation is a map $h$ with $h(g \cdot x) = g \cdot h(x)$. So $[G, \mathbf{Set}]$ is the category of $G$-sets. Non-example: a family of morphisms $\alpha_A \colon F(A) \to G(A)$ that fails one naturality square is not a morphism of $[\mathcal{C}, \mathcal{D}]$. A morphism of $[\mathcal{C}, \mathcal{D}]$ is an isomorphism exactly when all its components are isomorphisms.