Relation between Direct Sum and Hom

This lecture explores the isomorphism between Hom(A, B) and Hom(A, B) for any abelian group B, based on the proposition that for a given set X and {Ax | x ∈ X} in Ab, there exists a clear relationship. The proof involves showing the isomorphism between Hom(A, B) and Hom(A, B) through a specific mapping. The lecture also delves into the underlying sets of abelian groups and the importance of demonstrating specific relationships within them, leading to a deeper understanding of the connection between direct sums and Hom functors.