This research project aims to develop a computer-checked library of category theory within the Lean proof assistant, with a specific emphasis on concepts and examples relevant to functional programming. Category theory offers a robust mathematical framework that allows for the abstraction and comprehension of concepts across diverse fields, including computer science. Additionally, the project will explore the application of final coalgebras, particularly in the context of understanding infinite data structures. By creating a formalized category theory library within Lean, our objective is to enhance our understanding of functional programming and programming language concepts. Moreover, we aim to facilitate the study of these topics by providing a comprehensible library and a rigorous foundation for program reasoning, thereby benefiting researchers and practitioners in the field.