We examine a monoidal structure on the category of polynomial functors, defined through the operation of substituting one polynomial into another. We explain how this composition product transforms polynomials into a richer algebraic structure, enabling the modeling of more complex interactions and processes. The chapter explores the properties of this monoidal structure, how it relates to existing constructions in category theory, and its implications for understanding time evolution and dynamical behavior. We also provide examples and visual representations to clarify how substitution works in practice.

We formally define polynomial endofunctors on the category of sets, referring to them as polynomial functors or simply polynomials. These are constructed as sums of representable functors on the category of sets. We provide concrete examples of polynomials and highlight that the set of representable summands of a polynomial is isomorphic to the set obtained by evaluating the functor at the singleton set, which we term the positions of the polynomial. For each position, the elements of the representing set of the corresponding representable summand are called the directions. Beyond representables, we define three additional special classes of polynomials: constants, linear polynomials, and monomials. We close the chapter by offering three intuitive interpretations of positions and directions: as menus and options available to a decision-making agent, as roots and leaves of specific directed graphs called corolla forests, and as entries in two-cell spreadsheets we refer to as polyboxes.