In this video I define universal properties, universal morphisms, initial/terminal properties and initial/terminal morphisms. I illustrate how the product can be viewed as a universal morphism. I define an exponential object, and discuss how it corresponds to a universal morphism. I also define a natural number object, and describe how it corresponds to the set of natural numbers in the category of sets. I also define comma categories, and describe how universal morphisms correspond to initial/terminal objects within them. I also show that the natural number object corresponds to a universal morphism using a new proof. The proof is given in the unlisted video at

https://youtu.be/h0kVqeZCdJc

Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube.