Harold Simmons.
An Introduction to Category Theory.
CUP. 2011
Category theory provides a general conceptual framework
that has proved fruitful in subjects as diverse as geometry, topology,
theoretical computer science and foundational mathematics.
Here is a friendly, easy-to-read textbook that explains the fundamentals
at a level suitable for newcomers to the subject.
Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as my a recommended text for a taught introductory course.