Category Theory For Beginners: Representable Functors

We use the running example of the natural number object as the universal dynamical system to illustrate how representable functors can be used to understand universal morphisms and universal properties. We discuss adjunctions, free and forgetful functors, copowering and the category of elements. Our central result describes how universal morphisms correspond with representable functors. We establish this result using the Yoneda lemma, and show that the result can be applied to gain perspective on the nature of adjoint functors. We apply our ideas by computing the left adjoint of the forgetful functor which goes from the category of dynamical systems to the category Set. Much more about these ideas can be found in the book Category Theory in Context By Emily Riehl https://people.math.rochester.edu/faculty/doug/otherpapers/Riehl-CTC.pdf Also see my videos Category Theory For Beginners: Universal Properties https://www.youtube.com/watch?v=V9tMzmlpuYo&list=PLCTMeyjMKRkoS699U0OJ3ymr3r01sI08l&index=5 Category Theory For Beginners: Adjoint Functors https://www.youtube.com/watch?v=AppzvbDLxBw&list=PLCTMeyjMKRkoS699U0OJ3ymr3r01sI08l&index=13 Proofs related to this video can be found here: Result about Yoneda embedding https://youtu.be/obhRQ7SxOhc Proof that representability implies universality https://youtu.be/KffupXy9ByQ Proof that universality implies representability https://youtu.be/csEeQ_RatJI