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/fac...

Also see my videos

Category Theory For Beginners: Universal Properties
   • Category Theory For Beginners: Univer...  

Category Theory For Beginners: Adjoint Functors
   • Category Theory For Beginners: Adjoin...  

Proofs related to this video can be found here:

Result about Yoneda embedding
   • proof about Yoneda embedding  

Proof that representability implies universality
   • Proof that representability implies u...  

Proof that universality implies representability
   • Proof that universality implies repre...