Forgetful functors vs. properties

This lecture delves into the distinction between structure and property, exploring the realm of forgetful functors and their relationship with functor categories. Through various examples, the instructor illustrates the concepts of natural transformations, adjunctions, and the interplay between forgetful functors and inclusions of subcategories. The discussion also touches upon the injectivity and surjectivity of forgetful functors, prompting the audience to ponder the properties of their favorite forgetful functors. The lecture concludes with an analysis of functor categories, shedding light on the structure of Fun(D, C) and the set of natural transformations between functors.