Advancing categories with functors of functors

In this document, we review the basics of category theory and functors calculus, and we implement some novel concepts. In particular, we review the concepts of category, functors, and natural transformations. Furthermore we introduce the novel concept of func- tors of functors. We illustrate these concepts with diagrams to ameliorate the expression of these ideas. We conclude that these concepts open new avenues and are advancing category.