PlanetPhysics/Functor Categories

In order to define the concept of functor category, let us consider for any two categories $$\mathcal{\A}$$ and $$\mathcal{\A'}$$, the class $$M = [\mathcal{\A},\mathcal{\A'}]$$ of all covariant functors from $$\mathcal{\A}$$ to $$\mathcal{\A'}$$. For any two such functors $$F, K \in [\mathcal{\A}, \mathcal{\A'}]$$, $$ F: \mathcal{\A} \rightarrow \mathcal{\A'}$$ and $$ K: \mathcal{\A} \rightarrow \mathcal{\A'}$$, let us denote the class of all natural transformations from $$F$$ to $$K$$ by $$[F, K]$$. In the particular case when $$[F, K]$$ is a set one can still define for a small category $$\mathcal{\A}$$, the set $$Hom_{M }(F,K)$$. Thus, cf. p. 62 in, when $$\mathcal{\A}$$ is a small category the `class' $$[F, K]$$ of natural transformations from $$F$$ to $$K$$ may be viewed as a subclass of the cartesian product $$\prod_{A \in \mathcal{\A}}[F(A), K(A)]$$, and because the latter is a set  so is $$[F, K]$$ as well. Therefore, with the categorical law of composition of natural transformations of functors, and for $$\mathcal{\A}$$ being small, $$M = [\mathcal{\A},\mathcal{\A'}]$$ satisfies the conditions for the definition of a category, and it is in fact a functor category.

Remark : In the general case when $$\mathcal{\A}$$ is not small, the proper class $$M = [\mathcal{\A}, \mathcal{\A'}]$$ may be endowed with the structure of a supercategory (defined as any formal interpretation of ETAS) with the usual categorical composition law for natural transformations of functors. Similarly, one can construct a meta-category defined as the supercategory of all functor categories.