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 Hom_M(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.