Functor category $[\mathcal C,\mathcal D]$

Within the definable-class convention of Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{CAT}$ is not formed, the construction below is formed when the source category $\mathcal C$ is small. Then a functor out of $\mathcal C$ and a natural transformation between two such functors are set-coded data, so they can be objects and morphisms of a category in ZFC.

For categories $\mathcal C,\mathcal D$, the functor category $[\mathcal C,\mathcal D]$ has functors $\mathcal C\to\mathcal D$ as objects and natural transformations as morphisms (Natural transformation and its components). Its identities and composition are the identity transformations and vertical composition. These are the operations of Identity natural transformation and vertical composition.

Closure under composition is Vertical composites of natural transformations satisfy naturality. Associativity and the identity laws hold at each component because they hold in $\mathcal D$. Further smallness and local-smallness properties of this category are stated separately.

For an arbitrary large source $\mathcal C$, the same notation may be used only as metatheoretic shorthand for functors and natural transformations; this definition does not form those proper-class-sized data into a category.