Group Theory: Adjoint Functors and G-sets

This lecture delves into the adjunction between the 'free action' functor and the forgetful functor, connecting the category of sets and that of G-sets. The instructor explains the composition of usual applications, the preservation of composition, and the action of groups on sets. The lecture also covers the definition of the forgetful functor, the concept of G-equivariance, and the natural transformations between functors. Furthermore, it explores the structure of G-sets, the verification of equivariance, and the importance of natural transformations in preserving identities and compositions.