Definition 0.1. In order to define the concept of functor category, let us consider for any
two categories 𝒜 and 𝒜′, the class
of all covariant functors from 𝒜 to 𝒜′. For any two such functors F,K ∈ [𝒜,𝒜′], F : 𝒜→𝒜′
and K : 𝒜 → 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 𝒜,
the set HomM(F,K). Thus, cf. p. 62 in [1], when 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
∏
A∈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 𝒜 being small,
M = [𝒜,𝒜′] satisfies the conditions for the definition of a category, and it is in fact a functor
category.
Remark: In the general case when 𝒜 is not small, the proper class M = [𝒜,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.
References
[1] Mitchell, B.: 1965, Theory of Categories, Academic Press: London.
[2] Refs. [15], [17], [18] and [288] in the Bibliography of Category Theory and Algebraic
Topology. Categories, Functors and Automata Theory: A Novel Approach to Quantum
Automata through Algebraic-Topological Quantum Computations., Proceed. 4th Intl.
Congress LMPS, P. Suppes, Editor (August-Sept. 1971).
![M = [𝒜, 𝒜′]](https://images.physicslibrary.org/cache/objects/603/make4ht/FunctorCategories0x.png)